Racing drivers not NOT push as hard as they can as you describe. They push to certain limits, which are not just under/oversteer or until they crash, they drive to maintain the tyres (pushing too hard and destroying tyres is EASY) as well.
I wouldn't suggest that LFS's tyre heating code is perfect, far from it. But it's not as bad as you describe, and it is fairly easy to go quickly and look after your tyres if you set the car up right and drive it correctly - see the various endurance races for evidence. If you use an unrealistic hotlap setup (real life rarely has anything so extreme, even in hillclimbing) that is meant to work the tyres very hard (they only have to do one lap) then that is the problem.
Drive like a real racer and the tyres last. If anything they heat up too slowly when used in that manner.