RAW, there is no limitation.
As usual, this is not something that matters to the set of rules, so Aarakocra can play any instrument as well as any other person.
If you are looking for an instrument that suits them well, I would like something small / light (that is, it does not interfere with the flight) and that is based on strings, like a small harp or a lyre, since its claws would be excellent to choose the guitar.
If we go further, we might even think about altering the less suitable instruments to improve the user experience of aarakocran. So imagine a flute that would have a mouthpiece with very specific shape and finger holes to accommodate the beak and claws.
As for the question of not needing a flute, I, as a human, can whistle and clap and sing (not very well, but still) and, nevertheless, several musical instruments were a rather early invention of our history.