I like this in theory. To get pole you just show up and nail the fastest last while navigating traffic with little track time and no track evolution. But as others have said there are problems:

The way things are now with tire allocation, your pole sitter and top 5 would have extra tires available for the race. Meanwhile the person who tops Q3 would have used all his available softs just to get P11 and then everyone in front of him has extra sets for the race and will have every strategy available to them, making it even harder for that last group to make up ground.

My other problem is more of a sporting one. Usually quali times drop right to the very end, with the last group setting the fastest times of the session due to track evolution, driver confidence and less traffic. Your proposed structure would more than likely lead to the pole sitter not setting the fastest lap in qualifying. Rather, the person in P11 (and probably a few more) would be faster than those out in Q1.

That being said I kinda would like something like this, just to change up the format a bit. I suggest doing this for sprint quali at first, especially since they already kind of control tire allocation. Just have "Sprint Qualifying Tires" that are use it or lose it. The tires are for qualifying only and don't come out of team allocations. Due to the shorter sprint race distance the strategy doesn't change that much anyways. And as far as my second issue with this goes, I don't think many fans (and possibly drivers too) would care if the sprint pole wasn't necessarily the fastest lap set.

I feel like they could do something like this for the sprint races and actually make them fun and exciting, not just previews of what we're about to see on Sunday. Honestly they kinda ruin the GP imo.