Does the number of chimes collected as Bard count towards the mastery system?

If it doesn't directly (which it should) it does indirectly anyway since you directly scale form them, and thus you will perform more effectively, even in terms of just damage if you do.

Probably not, nobody knows really.

Wouldn't really be fair if it did, they spawn randomly so its not really up to you if you can always reach them.

But like Chaddy mentioned, they can affect your grade indirectly due to the stats they give you, allowing you to be more effective at influencing the outcome of the game.