How accurate is Nietzsche's account of the birth of tragedy?

If the question is indeed not about the part in question that's bracketed out, then well, focusing solely on

tragedy drama, as a practice, emerged from an originally solely-musical chorus. First people sang story-songs, especially dithyrambs, and then later came dialog and characters as the subjects of the songs.

then this is just an observational truism. As Aristotle already wrote in his Poetics (Poetics IV, 1449a 10–15), this just was the progression of forms. Nietzche did not innovate much about this 'theory' in his analysis.

First was just the singing chorus, then a special actor in dialogue with the chorus, then a second actor who could interact with both, and so on.

Stopped on his way to the tragic competitions of the City Dionysia and asked where he was headed, the average classical Athenian in the street would very probably have replied, ‘‘es chorous’’ (‘‘to the choruses’’) or perhaps ‘‘es tragoˆidous’’ (‘‘to the tragos- singers’’). If he was quizzed further as to just how he would categorize this activity so beloved of his city, the term mousikeˆ (the ‘‘craft of the Muses’’ and the origin of our ‘‘music’’) would have very soon entered the conversation. For the Athenians, tragedy was – fundamentally, predominantly, and persistently – a musical event.

Fundamentally, since – at least until the middle of the fifth century – tragedy was oriented around its singing-and-dancing heart, the chorus (choros), and (to dip our toes in the dangerous waters of origins) it was probably in the bifurcation of a single lead singer from a choral group that the distinctive double form of tragedy, with its individual actors and singing-dancing chorus, arose.

Predominantly, because the chorus, with its highly choreographed dance-songs performed by some twelve or fifteen elaborately costumed and masked men in the wide open space of the orchestra, must have been the dominant physical and aesthetic presence in tragedy, one that, unlike the actors, virtually never left the space that they entered at or near the start of a play. And even as the quantitative contribution of the chorus progressively declined over the course of the classical period, as though to counterbalance this musical loss, the actors began to sing more and more from the stage.

And persistently, because even centuries after the end of the most creative period of classical tragedy, audiences from Syracuse to Abdera and beyond clamored for the songs of Euripides as much as the speeches, to the extent that the craze could be deemed a national epidemic (see the fascinating story told by Lucian, How to Write History 1; cf. Axionicus fr. 3 PCG).
_{–– Peter Wilson: "Music" in Companion to Greek Tragedy, 2005}

If we just stop short at checking whether that part from Nietzsche holds up, then yes that's what I heard when I read Greek tragedy as well. Perhaps a bit more refined in that there where traveling poets/singers like Homer or Trepander, professionals who later attended regular musical festivals.

Nietzsche's account did face criticism, for other parts, especially for his speculation for example on 'originality' (~generative birth of genius) and the assumed focus solely on Dionysus and his cult. All in all it his deviance from the established observations into metaphysical speculation of 'true' origins that is seldom proven wrong, but also not proven right.

Another point of contention is that Aristotle and quite some authors since describe a progressive development towards perfection whereas Nietzsche's criticism focusses on a downward destructive trend, brought about by the influence of analytical philosophers. Aischylos is still OK, but all epigones get worse all the time, beginning with Sophokles. The rising emphasis on vocal dialogue is a diversion from the musical sense, purpose and raison of the whole spectacle. It is hard not read Nietzsche on that part as if he was writing also some kind of science fiction in the sense of cultural criticism of his own time.

_{–– Jochen Schmidt: "Kommentar zu Nietzsches. Die Geburt der Tragödie", in: Heidelberger Akademie der Wissenschaften (Eds): "Historischer und kritischer Kommentar zu Friedrich Nietzsches Werken", de Gruyter: Berlin, Boston, 2012.}

A proportion of these visitors to Sparta had come to perform at the contests or train the choruses. A scholion on the Alcman papyrus notes that the Spartans in the time of its author used foreign poets or chorus trainers.79 This is likely to have also been the case in the archaic and classical periods. It has been suggested that Ibycus’ Spartan ode was an early epinician offered to a Laconian victor at a contest in Sicyon. If so, Sparta may have been yet another of the cities visited by this peripatetic poet. A dithyramb by Bacchylides (20 Maehler) was probably composed for a Spartan chorus in the early fifth century. The papyrus preserves the title ‘Idas for the Spartans’ (Ἴδας Λακεδαιμονίοις). Idas is associated with the abduction of Marpessa from a maiden chorus. As Maehler has noted, such a theme would be suitable for performance by a chorus of young girls at Sparta, especially given that the foundation myths of many of the Laconian cults, including that of Artemis at Caryae, featured stories of rape.

As a vibrant centre for poetry, Sparta was inevitably linked to the network of festivals in the Aegean and Asia Minor. Sparta’s strongest connection was with the Aegean island of Lesbos. The Lesbian singer Terpander was known as the first victor of the Spartan Carnea, which may have been founded in the first quarter of the seventh century. To later authors Terpander was one of the first of a series of foreign poets, who would not only develop Sparta as a centre for choral poetry, but also enhance the stability of its constitution.
_{–– Edmund Stewart: "Greek Tragedy on the Move. The Birth of a Panhellenic Art Form c.500–300 BC", Oxford university Press: Oxford New York, 2017.}

A harsh criticism to both Aristotle – and by extension Nietzsche – illustrates our lack of 'real' historical evidence, as we know mainly the full-fledged classics of the, well, classical era, and at best fragments from the archaic era, so that we all have to rely on interpolations and interpretations:

These observations raise issues that go beyond the religious aspect of the origins of tragedy, which is our topic here, but it is important for our purposes not to misjudge or judge arbitrarily the role of festivals of Dionysus in the speculations about the origins of tragedy of Aristotle and his successors. The occasion of a festival event need not have any particular significance for its nature. The role of the Dionysia festival in Aristotle’s history of tragedy does not make tragedy Dionysiac in the modern sense any more than the role of Gorgias in Aristotle’s history of rhetoric means that he regarded rhetoric as an essentially Sicilian art. So too the advocates of primeval Attic tragedy were inventing a history that culminated comprehensibly in the City Dionysia, not presenting tragedy as unthinkable apart from Dionysus. No ancient writer speaks of tragedy as Dionysiac in the strong sense employed by modern scholars.

Aristotle knew very little about the origins of tragedy, and his successors, though bold in invention, knew no more. We can analyze their methods and contextualize or deconstruct their claims, but cannot know any more than they did. There is every reason to accept as sound Aristotle’s view that epic was the key influence on the development of tragedy. We ought not to father on him the view that tragedy is a manifestation of the Dionysiac spirit and inconceivable without it; that view is nowhere expressed or entailed in the ancient discussion of the origin of tragedy. Modern scholars offer various other grounds for regarding tragedy as Dionysiac, but these lose much of their force if one concludes that the standard modern view is not in, but has been read into, Aristotle.

Tragedies were performed at festivals on holidays, which were by origin holy days and certainly retained cultic components but, to judge from some of the plays, were certainly not occasions for the suppression of religious controversy and doubt. The question in what sense, if in any, tragedy is a religious phenomenon ought to prompt examination and debate rather than assumption and assertion.
_{–– Scott Scullion: "Tragedy and Religion: The Problem of Origins", in: Justina Gregory (Ed): "A Companion to Greek Tragedy", Blackwell: Malden, Oxford, 2005.}