It wouldn't lead to extra reviews. Teams should be allowed to review it and if they lose, it counts against their allowable reviews/timeouts.
The entire reason for the reviews is to just get it right. This is not a judgement call like forward progress being stopped. It's an actual fact of time has run out, play is over. Turning time into a judgement call is absurd. You're correct, there should be a buzzer that goes off in the officials ear. Otherwise, they may not hear it.