Why did it take so long to abandon sail after steamships were demonstrated?

Maybe it was that most, if not all, of then current ships were sailing ships to start with, and using the steam engine at first was thought to be an added on means of propulsion. However, that apparently changed later as the steam engine became more reliable as well as the availability of the fuel. A big downside of sails was that there were numerous places on the globe where the wind was not consistent or best used for sailing. Reliable steam engines, with enough fuel, negated the use of sails.

This question also seems about why it took steamships close to a century to take over, while it took oilships about 15 years. This has to do with an ever decreasing Product Life Cycle

The PLC phenomenon is that newer products in the same function will have a shorter economically viable life span. Think about how we had radio for a long time, this was replaced by B&W tv's which were replaced by colour TV's. These CRT's are now replaced by flat screens. These products all have the same function: Home Family entertainment in the evening.

You can also think about transistor radios => boom box => walkman, Mp3 player, iPod, smart phones. (function is portable music)

So, we can argue that the same principles apply to sail-steam-motor.

What's more, I'll hazard a guess that technically, going from a coal supply infrastructure to a an oil supply infrastructure is easier than building an entire coal supply infrastructure. This answer should be seen as an addition to the ones above.

Logistics and costs. Wind is free and so coal plus fresh water (as steamships use both) has to preform better than free wind. Both are heavy and take away cargo space.

The US Great White Fleet sailed around the worlds. This shows the logistics of coal.

The ships of the Great White Fleet were no exception. In that era, a battleship steaming at sea speed consumed its coal supply within a week Fresh water— crucial throughout maritime history—was even more important in the age of steam power, since steamships were dependent on liberal amounts of fresh water to resupply their boilers. Then there was the question of feeding warships’ crews. On the voyage of the Great White Fleet, the crew complement of the fleet consisted of some fourteen thousand men. ,,,

and

A study the Naval War College conducted in early 1907 estimated that the Great White Fleet would require some one hundred chartered colliers to support it on its voyage around the world.

https://digital-commons.usnwc.edu/nwc-review/vol71/iss4/6/

And it's not so different now in the days of oil/turbines. The Royal New Zealand Navy has two surface warships and 1 tanker. The Royal Australian Navy has 11 warships and two massive tankers.

Cost and range.

Coal is not cheap (in money or lives, in the early industrial area), and you either need a network of coaling stations, or a large ship to carry enough coal and still have room for payload. This meant that, until the use of iron in the 1840s broke the barriers on ship size imposed by wooden construction, steam wasn't a viable option for Atlantic crossings (though it had a role as auxiliary power and, via steam tugs, for port entry and departure)

In contrast, the limit on a sailing ship's range is the ned to carry food and water for the crew - crossing the Atlantic was pushing the capabilities of sailing ships 350 years earlier, and they quickly evolved to handle longer passages.

Coaling stations are worth examining in a bit more detail. They aren't much use without a supply of coal! Unless you can mine it locally, you now need to transport enough coal to refuel your steamships, as well as the actual payload.

Unless you make use of another power source to transport the (lower value and less time critical) coal, you are left with something resembling Tsiolkowski's rocket equation, to calculate the increasingly vast amounts of coal required to replenish the network of coaling stations to keep the steamships running.

So, naturally the colliers were sailing ships.

Thus steam actually guaranteed a role for sail for a substantial part of a century - and left sail as the method of choice for bulk cargo, pretty much into the start of the marine diesel era.

Range and reliability were primary factors.

The earliest steam powered ships used atmospheric Watt style engines, those that condensed hot steam to create a partial vacuum to create power. Not very efficient.

Consequently, the early steam ships had a fairly short range... the first steam ship to cross the Atlantic, the SS Savannah, did so largely on sail power. There wasn't room to carry enough fuel to make the trip entirely on steam power.

The much more efficient pressure steam engines didn't come into use until metallurgy had advanced to the point that steam boilers and joints could be built that would take the pressure and heat. The early steam ships used somewhat inefficient paddle wheels, as opposed to the more efficient screw propeller developed later.

Also, early steam engines were prone to frequent breakdowns. Lack of experience with designing and operating large steam engines, and lack of metals that could take the stress were contributing factors.

On short trips, and especially on inland waterways such as rivers, that were narrow and lacking the constant wind of the open seas, even the early inefficient steam ships were considered a great improvement due to their ability to move regardless of the wind.

1- Reliability. The first steam engines were prone to breakdown. That's not particularly nice in a mine or in a factory but outright dangerous on a ship. It took time before engines were reliable enough for ocean journeys.

2- Sufficient range. You can't bunker coal in the middle of the Atlantic. Ships need sufficient bunker capacity to cross an ocean and still be able to operate economically. In other words, ships had to be large enough to carry both coal, cargo and make a profit. When the first steam engines appeared, ships lacked the capacity for both. That also took time.

3- Infrastructure. One harbor on each end of the Atlantic with enough bunker capacity is not enough. You need coaling stations everywhere. And again, it took time to develop a workable coaling network.

Developing all of the above took time.

The premise "If steam was superior to sail..." needs to be properly examined. Certainly steam was superior overall (once the technology was mature), but there were several areas where sail had the edge up to the end of the 19th century.

• Speed in favourable conditions. As others have mentioned, no steam ship could beat a tea clipper from China to Europe, even without counting the refuelling stops. It was not common to have a predictable route, periodic rather than constant, where speed was of the essence; but in such a case, steam would be noticeably worse.
• Emergencies. Being caught in the Doldrums could immobilise a sailing ship for weeks; but a boiler explosion (analogous to the 'blue screen of death' that incapacitated more than one US Navy ship when computers were going from useful to ubiquitous) would immobilise a steamship until it was towed into port or abandoned.
• Cost. Before internal combustion, and away from the limited railways, the only alternative to sea transport was a horse and cart. A coaster taking a load of bricks sixty miles up the coast, loitering until a cargo of grain was ready to go to the city, and then bargaining to transport some heavy machinery for a farm, could make a good profit; but not if every mile covered, full or empty, needed coal. Indeed, naval architects are still looking at putting sails on container ships to reduce fuel bills (though it's fair to say nobody has produced a workable design).
• Capacity (similar to but not the same as the above). Coal bunkers took up a fair proportion of any cargo ship, reducing the useful load. This was the reason that many naval colliers (taking fuel from the source to a dockyard where the fleet refuelled) used sail; if you are replenishing a huge stock, large amounts at slightly unpredictable intervals are better than small amounts every Tuesday afternoon.

All of these are rare cases, but help to explain why sail lasted so long in a few corners of the industrialised world.

Evolution is slow

You have fleets over fleets of sailing ships, and then someone, somewhere begins to build steamers. First, the steamer-wharf capacity needs time to ramp-up; then, the steamers need to gain superiority over the sailing ships; and finally the sailing ships need to be phased out. You simply just don't scrap sailing ships just because steamers are available.

For comparison, even in World War 2 plenty of horses were used for various tasks. And WW2 is several decades after the motor-car had been invented.

Even when the infrastructure was in place, why abandon a sunk asset which can still produce some revenue?

Eric Newby wrote The Last Great Grain Race about his 1938 voyage as crew from Port Lincoln, South Australia to Glasgow, Scotland. These tall ships were carrying wheat grown on the plains of South Australia to the UK market. It did not matter to the wheat that the voyage was slower and less comfortable than a steamship. It was cheaper, which mattered to the purchasers of the wheat.

Conversely, migrants from the UK to Australia would book a steamship berth. The South Australian Maritime Museum has a hands-on display of the berths of the sailing era, the early steamship era, and the later liner era. Each has a substantial rise in comfort: from a hay-filled mattress shared with others, to a dormitory room of 6-8, to a twin-share room. The displays also note the fall in voyage duration; and the fall in mortality with each era (especially as the abundant power available in steamships allowed frozen food to be kept refrigerated).

I realise this answer is about the final years of sail. But I hope you find it useful to know how those last years were played out in far-away South Australia, home of last of the working sail ports.

You can't completely replace sail with coal until you are 100% sure that you are going to have access to coal everywhere you need to go.

This is basically an extension of Steve Bird's #1 and #2. It's beyond the economics and into the availability.

Do you have reliably supplied coaling stations all the way to, say, Australia? If it's a military operation, will all the ports be friendly? If not you need to keep sails on hand.

As an example the famous Clipper Ship Cutty Sark was one of the fastest trading ships in the world when she launched (1869) and was used for the China tea trade. But coal quickly covered that route so then she was diverted to trade with Australia.