This needs a frame challenge and then still ends up being some kind of a list answer, if answered just straight.

The very first frame challenge would of course be that the very word for the concept is of fairly recent origin:

celebrity 4. concr. A person of celebrity; a celebrated person: a public character. 1849: Miss Mulock Ogilvies ii, “Did you see any of those `celebrities,' as you call them?”

— OED 2nd edition

Although what gained currency then with already our word for it in the same sense as most would define this today, other instances might be found in quite some recognisable earlier examples, even if that was expressed in different words.

What is a celebrity, and what might resemble one in the specified timeframe? Today 'celebrities' are often just 'celebrated' for 'being celebrities'. That is famous for being famous, or just infamous, not really what they do, because they do essentially nothing of any relevance (with the exception of perhaps one —possibly very small— act of any kind to grab the attention).

But as the question clarifies:

Q: looking for the sorts of people who are famous for what they can do, not for who their parents or family were.

This is in its essences certainly much broader.

With the modern concept of 'celebrity' reduced to 'being famous', that is 'known throughout the country' without people 'recognising such names' being personally acquainted with the person, we get an endless list of people.

So, which people would be ultimately widely known despite not being somehow princes, royalty or the like?

People like gladiators, charioteers, poets, actors, even who*es spring to mind easily. And from there we see that so do scientists, politicians and preachers and criminals.

Dante, Shakespeare, Goethe, or the likes of Robespierre, Luther, some famous highwaymen or Sacheverell?

Sacheverell?

This was the first example given in a recent book on the topic:

[…] Doctor Henry Sacheverell, and he was an ambitious Anglican minister who gave a fiery speech from the pulpit of St Paul’s Cathedral on 5 November 1709, attacking ‘false brethren’ in the church. Delivered on the anniversary of Guy Fawkes’s plot to blow up King James I and his Parliament, it was immediately perceived as a political broadside against the Whig party by pro-royalist Tories, and thus Sacheverell was thrust into the centre of a political maelstrom. And, in case you’re not up on your maelstroms, the eighteenth-century public loved ’em!

Before long, 100,000 copies of his speech had been sold, with estimates suggesting as many as 250,000 people may have encountered the text. London’s population was probably no larger than 600,000 people. His next speech was attended by a huge mob, desperate to get into the church. Alarmed at his sudden popularity, Queen Anne’s ministers in the Whig government attempted to try him for sedition; but he’d selected his words too carefully, so they instead successfully charged him with High Crimes and Misdemeanours in the House of Lords. […]

Another obvious comparison was Edward Teach – better known as Blackbeard the Pirate – whose ingenious instincts for celebrity branding saw him stuff burning fuses into his thick black beard. This was all part of his performance as deranged slaughterer, and he undoubtedly spilled blood along the way – there’s no such thing as pacifist piracy – but he was a cunning tactician who much preferred intimidation over violence; why risk a dangerous sea battle when he could terrify enemy sailors into surrendering without a fight? After a two-year spell rampaging around the Caribbean and making a nuisance of himself with American coastal towns,* he was hunted down in 1718 by the coastguard and, after an exciting battle worthy of a Hollywood popcornfest, had his head sliced off by a British lieutenant. The head was apparently hoisted up on the ship’s rigging and then was displayed back on the mainland, with legend telling that someone fetched it down, plated it in silver, and turned his skull into a gruesome tankard used in a Virginia tavern. Blackbeard lived a celebrity and died as one too.

— Greg Jenner: "Dead Famous: An Unexpected History of Celebrity from Bronze Age to Silver Screen", Weidenfeld & Nicolson: London, 2020.

Another example might be found in

George Bryan "Beau" Brummell (7 June 1778 – 30 March 1840)2 was an important figure in Regency England and, for many years, the arbiter of men's fashion. At one time, he was a close friend of the Prince Regent, the future King George IV, but after the two quarrelled and Brummell got into debt, he had to take refuge in France. Eventually, he died shabby and insane in Caen.
Brummell was remembered afterwards as the preeminent example of the dandy, and a whole literature was founded upon his manner and witty sayings, which have persisted until today. His name is still associated with style and good looks and has been given to a variety of modern products to suggest their high quality.

The definition problem might come down to discover congruency of the modern 'celebrity' concept with earlier phenomena, some theoretical deliberations:

Famous for being famous

[…] it turns out there’s no single accepted definition of celebrity among sociologists and historians. This isn’t very helpful. […] ‘Celebrities? Bah! Bunch of useless nobodies, most of them just famous for being famous!’ […]

[…] political theorist Daniel J. Boorstin, who defined a celebrity as ‘a person who is well-known for his well-knownness’. […]

[…] but Boorstin’s celebrity cynicism presents a false dichotomy. This stuff is way more complex than simply pitting supermodel hottie against Renaissance poet. Firstly, noble repute can exist both inside and outside of celebrity. Not all famous people are celebrities, but not all celebrities are passing fads. The Victorian poet Alfred, Lord Tennyson was hailed as a literary ‘lion’ in his lifetime, not because of his majestic mane (though he did sport luxuriant hair, he was more of a stone-cold fox in his youth but because lionism was a specific type of intellectual celebrity doled out by elite society to a gaggle of serious, chin-stroking brainboxes. This subset of celebrities were still chased around by excited fans, but their fandom was limited to a privileged class with access to high culture. In particular, lions were the sorts of men (it was typically men) who were guests of honour at salons hosted by aristocratic women.

Today, the same still applies. Boorstin decried the cluttering crap of lowbrow pop culture which devalues deserved fame, but such greatness isn’t extinct; there are still brilliantly talented celebrities whose contributions bring joy and cultural enrichment to millions, and whose influence helps redefine gender norms, sexual mores, political ideologies, and the boundaries of good taste. For all the supposedly inane, fake-tanned triffids invading our culture with their stupefying superficiality, and ricocheting between TV reality shows and Instagram adverts for tooth-whitening kits, there are also celebs whose statuses were earned through genuine ability and hard graft, and who contribute to the health of society just as much as some long-dead Austrian composer. […]

Meanwhile, Casanova also documented the fluctuating reputations of other gossip magnets, including Kitty Fisher, whose fame as London’s hottest courtesan was earned in the beds of her wealthy lovers. Her celebrity was heightened in 1759 by her falling from a horse while, as was the custom at the time, wearing no knickers – yes, she enjoyed a tactically brilliant ‘wardrobe malfunction’ that saw her dominate the gossip sheets for months thereafter – and she asserted her visual iconography by working closely with the portrait artist Joshua Reynolds. […]

[…] Boorstin’s book is an energetic polemic, but he was fundamentally wrong. Not only is celebrity three centuries old, but the ambition to be famous for the sake of being famous was already thriving in the age of Mozart. Indeed, even in 1786, shortly after Ann Hatton published her first poems, Thomas Busby published a book called The Age of Genius which bitterly complained how a new infestation of shallow celebrities was ruining traditional fame. Boorstin was 200 years late to the party. However, in his defence, I can see how he tied himself in knots, because celebrity isn’t always easily distinguishable from other types of public reputation. What Boorstin was championing wasn’t fame, but what I prefer to call renown.

— Jenner

That also some scientists shared in this famousness, renown, popularity might be demonstrated in examples like lowly background/from a refugee family optician becoming known throughout Europe for being the first producing commercially viable achromatic lenses, chiefly useful for much better telescopes: John Dollond.

That this might be criticised within the assumed frame for this question as perhaps applicable to middle and higher classes, well-to-do, other scientists seems obvious, but at once points to a general problem for answering this question: while we might assume that at least literate classes even were interested in what these 'celebrities' achieved, we also suffer from the fact that on the other hand a living remotely peasant might indeed be fully unaware of such people's very existence, the illiterate and lowest classes are of course seldom any kind of study object for this kind of social history. We simply do have much more written stuff to know about from those people who could write…

So relying on direct first hand accounts is bound and restricted to this kind of educational level, and reliance on produce of the printing press suffers similarly: namely that apart from mass media being a boon to any kind of (un-)popularity, all word-of-mouth 'talk of the town' kinds of information can only be gleaned indirectly from these types of sources.

However, some scientists were well known by lots of people we would think of rather disinterested otherwise regarding scientific progress: they went very public with their practical experiments. Sometimes leaving us their names as words for certain qualities or activities: like Franz Mesmer.

Let's raise the dead.

On January 18, 1803, Giovanni Aldini, Professor of Physics at the University of Bologna, conducted a series of galvanic experiments on the body of George Forster at the Royal College of Surgeons in London. […] After Galvani’s death in 1798, Aldini started travelling across Europe, electrifying the bodies of dead animals in public demonstrations staged as theatrical spectacles. After he conducted one such experiment before members of the Royal College of Surgeons in London, involving the galvanic reanimation of the decapitated head of a dog, Aldini received several professional requests to reproduce this experiment, this time using a human body. […]

Like Aldini, then, Ure was not a marginal or eccentric figure within his scientific community, but rather highly esteemed and respected. His experiment was conducted on November 4, 1818. He delivered a report of this to the Glasgow Literary Society on December 10, 1818, which was published in The Quarterly Journal of Science, Literature and the Arts the following month. Like Aldini, Ure used a combination of public spectacles, professional lectures, and press reports to maximise the impact of, and recognition for, his work. […]

This grim spectacle, in which a dead body was jerked around like a puppet by a new scientific technology before the horrified and astonished eyes of an audience, captured the popular imaginary in the 19th century. While the immediate audience for Aldini’s attempt to reanimate Forster’s body was a small one made up of medical and scientific men, the sensational reporting of the case in the popular press meant that news of the event also achieved broad circulation.

— Elizabeth Stephens: “Dead Eyes Open”: The Role Of Experiments In Galvanic Reanimation In Nineteenth-Century Popular Culture", Leonardo, Vol. 48, No. 3, pp. 276–277, 2015. doi

For more of this problem on definitions used or even usable, together with a discussion of possible bounds and frames with examples:

Who is England’s first celebrity? Edward Alleyn? Henry Sacheverell? Jack Sheppard? David Garrick? Joshua Reynolds? Mary Robinson? Lord Byron?

While origins may be undiscoverable as well as theoretically untenable, the question’s lure remains, and it hovers in the background of many 17th- through 19th-century studies that focus on popular individuals and employ the word ‘celebrity’ in title or text. Beyond the critical longing to establish definitively the ‘first’ of something, this question is complicated by what we mean by the word celebrity, especially in earlier periods. Pre-19th-century writers, of course, did not use it in its modern understanding. How do we construct a definition of celebrity that doesn’t primarily serve to justify the origins selected or the destination reached? This article will be not only present some of the definitions and examples current in the burgeoning field of what might be called ‘early celebrity studies’ – as well as identifying some contenders for first celebrity – but it will also sketch the broader interdisciplinary research necessary for such study.

— Cheryl Wanko: "Celebrity Studies in the Long Eighteenth Century: An Interdisciplinary Overview", Literature Compass 8/6, pp351–362, 2011. doi

On that primary example the journalist Wilkes (museum artefact: satirical print) we see:

The print shops of mid-century London teemed with more attractive options. Ample watch-papers framed the city’s top courtesans and loveliest duchesses in decorative hearts, underscoring their ineffable sway over men’s souls. Yet many a Londoner must indeed have cut carefully around the circle at the heart of the broadsheet, then transferred the unlikely portrait of John Wilkes to his own timepiece—gazing with admiration on a visage so ugly its owner was fond of saying he needed “half an hour to talk his face away.” In 1760s London, Wilkes perched at the absolute summit of his popular appeal, and London was obsessed. “Little is talked of but Wilkes, and what relates to him,” wrote an exhausted Horace Walpole in late 1763; “[he] is reverenced as a saint by the mob, and if he dies, I suppose people will squint themselves into convulsions at his tomb, in honour of his memory.” Wilkes had been unjustly arrested by the Crown months before, inaugurating a legal case that riveted the nation.

— Kevin Bourque: "Cultural Currency: Chrysal, or The Adventures of a Guinea, and the Material Shape of Eighteenth-Century Celebrity", in: Ileana Baird & And Christina Ionescu (eds): "Eighteenth-Century thing theory in a global Context. From Consumerism to Celebrity Culture", Ashgate: Farnham, Burlington, 2013.

Apart from problems over the definition of celebrity, the timeframe inquired about seems to be phrased in two aspects not necessarily congruent to each other: restricting 'modernity' apparently with 'post-20th century' and inquiring about 17 to 19th century might fit with certain definitions of modernity in for example art history, but also clash with sometimes much broader sociological or historical definitions that might start with Renaissance or Early Modernity, very often with the Industrial or French Revolutions. Therefore, some historians might define both aspects as subsets differing over those definitions, with their own re-stricting of the meaning of 'celebrity' as a thoroughly 'modern' phenomenon in itself, although that periodisation favoured in most sociologically defined angles then would include still the 19th century frame of the question:

[…] the business of renown and celebrity has been in the making for two and a half centuries. […]

Renown, we shall say, was once assigned to men of high accomplishment in a handful of prominent and clearly defined roles. A sixteenth-century jurist, cleric, senior mercenary, or scholar was renowned for bringing honour to the office he occupied. He might be acclaimed in the street, but the recognition was of his accomplishment—his learning (in the case of John Donne, for instance), his victories (as Othello is acclaimed in the play), his implacable power (in the case of Cardinal Wolsey). Renown brought honour to the office not the individual, and public recognition was not so much of the man himself as of the significance of his actions for the society.

This historical difference is readily studied by way of the fame of one of the very few women of historical renown in the period before celebrity became a feature of the individualisation of fame.

This book offers an explanation of this no more than 250-year-old phenomenon. It finds the reasons for the persistence, the vigour, and the apparently limitless energy of the new spectacle and its peculiar allegory in its history. Mid-eighteenth-century London is our starting point, half a century after the capital replaced the court as the centre of social dynamics. Spontaneously, the city bred its version of a new social figure, famous for his and her urban accomplishments: Dr. Johnson and his self-appointed circle of public opinion–makers in literary journalism; Wilkes and his raffish radicals; Lady Mary Wortley Montagu and the lead she took as a solitary woman tourist (getting into the Sofia mosque disguised as a man), as philanthropic proselytiser for the new science of immunisation (herself disfigured by smallpox), as friend of poets (Alexander Pope) and audaciously free-loving free-liver; the amazing Joshua Reynolds, all were treated as first, authentic celebrities. Above all, the late eighteenth and early nineteenth centuries provide the earliest opportunity to study the way in which the theatre, distorting and magnifying mirror of its society, assumes the significance it never loses as providing the leading ladies and men of the cast of celebrities. Sarah Siddons, David Garrick, and Kean anticipate Bernhardt, Ellen Terry, and Irving and point forward to Hollywood. […]

The soberly named Literary Club, however, of which the first advocate was Joshua Reynolds, was a very much more portentous, serious-minded, and public affair. Conscientiously sober, dressed in the new black of the gentleman scholar, they embodied perfect politeness in its new sense.

Reynolds and eight other members began the club at the high peak of London’s clubbishness in 1764. The names of the members it accrued over the thirty years of its incarnation vindicate its claims to both seriousness and influence. Reynolds himself, Edmund Burke, Oliver Goldsmith, James Boswell (on his merits as well as Johnson’s ticket), Garrick, Sheridan, Gibbon, Adam Smith, even Charles James Fox, demagogue and radical MP, were the cream on an intellectually very rich jug of talent.

Talent was the point. These were or became men of high achievement, who were quite purposefully in search of fame and public recognition, the first formula of celebrity. The first nine members Johnson himself in a ringing roll call (modestly omitting himself ) marked out as describing the circle of the arts, the sciences, and their disciplines: “we have Reynolds for Painting, Goldsmith for Poetry, Nugent for Physics, Chamiere for Trade, Politics and all Money concerns; Mr Burke for Oratory, Mr Beauclerk for Polite Literature, Langton for Ecclesiastical History … Sir John Hawkins for Judicature and Ancient Musick.[…]

It is safe, however, to say that at this juncture, celebrity was inseparable from the public acknowledgement of achievement. These men attached the glitter of celebrity to the solidity of what they had accomplished by merely appearing in public; they were recognised for who they were as a result of what they had done. A genius, however, like Reynolds, could turn these simple recognitions into the manufacture of himself-as-a-famous- painter. It took prodigious hard work but that was nothing to him. He loved it, and he loved what it brought. […]

Joshua Reynolds pretty well single-handedly invented the role of painter as public celebrity, threw off by sheer industry the heavy encumbrance of the patron, devised an unprecedented harlequinade of social life, compounding sexual licence, gluttony and drunkenness, gaming and court appearances before the king, all along with a ruthless business acumen for later artists to copy and embellish. Since he also possessed the necessary talents to accompany the show, he provides a sumptuous object-lesson for our theme. Hogarth was, by contrast, an artisan who picked up status by a judicious marriage. To contemporary taste, his bawdy, caustic realism, heir to Brueghel and Jan Steen, may make for a keener response in the sensibility, but beside Reynolds, Hogarth cut a very small dash.

— Fred Inglis: "A Short History of Celebrity", Princeton University Press: Princeton, 2010.

Quite similar in the late diagnosis of this phenomenon, as Inglis analyses the theme for the US from a view that New York and Chicago would start to emulate examples and begin to offer their own developments compared to those in Europe, Douglas/McDonnell offer a very sociological perspective that defines 'celebrity' as:

While there have been, for millennia, people who rose to fame, the pro- duction and proliferation of celebrities in the United States really takes off in the mid-nineteenth and early twentieth centuries as a result of sev- eral intertwined phenomena: the increased democratization of society, the evolution of the United States into a much larger and more imper- sonal society, the rapid growth of national markets and cultures, the rise of an emerging middle class and urban, white-collar workers, the spread of popular entertainments, and the gradual rise of bureaucracies to manage businesses and the government. Increased urbanization, fueled in part by rising immigration, meant there were growing and concentrated audiences for popular culture fare. The success of the “penny press” in the 1830s and beyond, geared to everyday people and not just elites, provided venues for advertising concerts, performances, and other events. A new kind of public sphere was emerging as well, one that was more participatory and less governed primarily by white, property-owning men.

— Susan J. Douglas & Andrea McDonnell: "Celebrity. A History of Fame", New York University Press: New York, 2019.

How much some the type of sex workers — often called 'who*es' without much second thoughts over the terminology that has been re-appropriated to a large extent back to its original direct but more neutral meaning– became celebrities might be exemplified by:

The Adventures of a Guinea has since fallen into sad neglect, “dropped,” as Mark Blackwell has put it, “from even the most eccentric list of the period’s canonical works.” And criticism for the past hundred years or so has had some trouble in parsing the reasons for Johnstone’s considerable success. […]

the author of his Dictionary of Literary Biography entry laments “Johnstone’s failure to organize these elements into any sustained narrative form” or to “bind these sketches together into an effective whole exhibiting character-development.” Travelling from palm to palm in rapid-fire exchange—in the span of a single if memorable sentence, from pimp to who*e to bully to pawn-broker to beau to tavern-keeper to bank to, finally, “the first minister of state”—Chrysal is a “narrative of irresolution,” a text which barely initiates one plot to abandon it for another. […]

Readers of The Adventures of a Guinea in the 1760s immediately recognized, in the person described in the novel as “the most celebrated courtesan of the age,” the iconic Kitty Fisher—“the most pretty, extravagant, wicked little who*e that ever flourished.”

Having staged a fall from her horse in St. James’s Park, giving upon landing “favourable opportunity of viewing those charms which decency dictates should be hidden,” Fisher’s sexual daring, extravagance, and cupidity riveted London. She, too, was quite literally objectified. Mezzotint engravers issued her image in droves at mid-century, and consumers paid homage to her by buying Kitty Fisher songbooks or sporting Kitty Fisher bonnets. “There are prints of her everywhere,” sniffed Giustiana Wynne in 1759. “I don’t find her beautiful, but the English do and that is what matters.” — Bourque

This Kitty Fisher:

She was one of the world's first celebrities that was not famous for being an actress, musician, or member of the royalty, but simply for being famous.

is footnoted with:

Letter from Thomas Bowlby to Philip Gell (January 5, 1758), The Ninth Report of the Commission on Historical Manuscripts, 2 vols. (London: Eyre and Spottiswoode, 1884), 2:402. Catherine Maria Fisher (1741?–1767), the best-paid and most exclusive courtesan in London, appeared on the arms of many of Britain’s aristocratic lords and military heroes; she nearly became a countess several times over. Giacomo Casanova met her covered in thousands of pounds worth of diamonds, and in 1759, engravings of her, based on portraits by Joshua Reynolds, were so popular as to be inescapable. Fisher perished in 1767, either of consumption, smallpox, or lead poisoning from cosmetics, and her body, laid out in her best dress, was exhibited before internment.

— Kevin Bourque op cit