every time I read the title of this thread I read it as mutilation. 
Same!
To the original idea.. i respect your reasons and can see the benefits to your idea, but have to agree with some of the other folks here, it is essentially taking the gambling element out of the team's hands and putting it in the hands of fate.. the reckless cowboy that he is.
If you wanted to see each car's hot lap, maybe the FIA could activate in car footage for all 24 so that you can watch the lap from the seat of your favourite driver after Quali is done? Not sure if there would be any technical blocks here, i think in the race you can only ride on board with 2 or 3 at a time.. not sure if they would go to the effort for the few hardcore fans that would use it though..