And what happens if you wind up with king and rook vs king and rook?
Some positions simply do not allow for a win. Yes, you could say do it on time. But then it becomes about mechanical dexterity as people try to be faster than their opponent in a pointless piece shuffle.
The proposed rule change doesn't make sense, so I can't say what the ramifications would be. Charitably - you're now suggesting that others invent and propose a rule-change which makes draws significantly less likely.
There are no "on-board" rule changes you can make which won't destroy the game of chess. Any rule-changes have to be "meta" changes affecting points in a tournament, the ELO system, or ways to encourage players to play a wider variety of opponents. That's why everyone's talking about what it might look like to modify the points system in tournaments, because it's the most practical thing to actually change.
Some positions simply do not allow for a win. Yes, you could say do it on time. But then it becomes about mechanical dexterity as people try to be faster than their opponent in a pointless piece shuffle.