Actually, it does work if the international segments are not the same day.

I recently used it for LAS-LAX-BOS (a leg within a larger SDQ-BOS-LAX-LAS r/t). The definition of "day" may well be "within 24h" which would mess up some that just involve an overnight.

EDIT: Now why AA can't seem to offer online checkin for international flights is beyond me. CO does, CX does, pretty much all European airlines do. You can't even complete a kiosk checkin for international with AA without having to also see an agent.