Needless to say, no one likes this idea because it throws out of the window centuries of game theory. Endgames would be completely different. I'm not convinced it would be a less interesting game, though.
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.
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.
For example a King vs King endgame. Even really weak players will never accidentally put their king next to the opponent.
These are the same.
Getting rid of check does make for a better game at beginner levels.
It's both easier to teach and leads to exciting finishes ss noobs hang their king.
It depends on the locations of the kings relative to the pawn ( you generally want it in-front of your own pawn ), and the concept of opposition.
So essentially it's a lost position for one player, but due to stalemate rule it will count as a draw.
(Which affects some draws but not most.)