RAW, there is no limitation.
As usual, the rules do not hold to heart, so Aarakocra can play any instrument as well as anyone else.
If you are looking for a well-fitting instrument for them, I would prefer something small / light (that is, not annoying theft) and strings, such as a small harp or a lyre, as their talons would make 39, excellent picks.
If we go further, we could even think about modifying the less suitable instruments to improve the user experience of Aarakocran. Imagine a flute with a mouthpiece and finger holes very specific to accommodate the beak and greenhouses.
Regarding the question of not needing a flute – me, as a human being, I can hiss, applaud and sing (not very well, but still) and yet, various musical instruments were a rather recent invention of our history.