It could work but it would cost clubs and I wouldn’t like it

Huge board visible to all showing when the time is stopped. Tannoys won’t be good enough and you often can’t hear them
Extra staff - one independent time keeper and his assistant.
Resilience in the system
Training costs
An oversight body regularly changing what does and doesn’t constitute additional time.
Instead of one outcry at the end of the game there’d be regular outcry during the game as fans moan when the clock starts and stops. Imagine managers berating the fourth official?

On balance then No.