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Discord does something like what you describe. It's kind of awful for music(e.g. if it's a channel with a music bot) as you'll hear it speed up and slow down in an oscillating pattern. The same effect also appears in games if you should have a game loop that always tries to catch up to an ideal framerate by issuing more updates to match an average - the resulting oscillation as the game suddenly slows down and then jerks forward is hugely disruptive, so it's not really done this way in practice.

Oscillations are the main issue with "catch-ups" in synchronization, and dropping frames once your buffer is too far behind is often a more pleasant artifact. It's not really a one-size-fits-all engineering problem.