Hmm, I thought I had done a post on this some time ago. I remember thinking about it. Maybe I started writing a post and scrapped it (I do this quite often).
There's a lot of nice things you could do with enforced lap limits for tires. For a start, it would mean you could make them a lot stronger / durable. Races would become more of a sprint, if the tire lap limit was set just before they start to degrade under punishing use. You could also have interesting strategic options if all types of types were always available - eg super soft tires could have limit of 10% of the laps on some tracks (including qualifying) while hard tires could do 50% (or more). The lap limits could be adjusted both for safety and the show, rather than the tire designers coming up with something and then the teams fighting back to overcome it.
But, there's a number of problems. I think you'd have to have some leeway else you could have races where the entire field pits on the same lap, which would be very dangerous if it happened when they're close together. Maybe some way to trade a lap on one stint for another to mix things up. Another issue would that it would be blatantly artificial. Of course, deliberately bad tires are artificial (and potentially dangerous too) so it depends how you feel about the different styles - in a sense, there still is a tire war but now it's between the designers on the teams and the designers at the tire manufacturer.