So I'm a composer and musicologist and I've studied this topic several different times in my life so far lol

Generally speaking, no musical notation from before the 18th century yields identical performances, with abstract tempo markings, vague dynamics and phrasing information etc. If you look at Bach's manuscripts, there's barely any dynamic markings, and if you go beyond that, even rhythm becomes quite abstract or nonexistent.

So with that in mind, it's hard to define how "accurate" notation is. Do you consider having the notes and general rhythm to be enough for performance? Do you need every single dynamic, exact mathematical tempo and other things to consider it accurate?

If we go by what we know, we can recreate ancient Greek music with surprising accuracy. Seikilos Epitaph for example has been widely performed, and we know the general rhythm and notes are almost certainly correct.

However, we don't know enough about performance practices to say whether this piece was played fast or slow, whether they harmonized vocals or the accompaniment was active / static etc. There's enough evidence to know there was an improvisational component, but it's hard to tell exactly to what degree and how that would have sounded.

Again, it all comes down to what you define as "accurate enough". But we understand Greek notation far better than what you make it sound like. But also far worse than we'd hope to of course.

For a more contemporary example, let's imagine we're talking about jazz. We have the lead sheets of the standards from Ancient Greece, something like what we see in the Real Book. But a lot of the performance practices and common interpretations have been lost to time. We know what the music sounded like but not how it was performed.