Hmmmm - it would also make quali take about a decade. There are 22 cars in the field. You are planning to eliminate the slowest on each run. So that is 21 full pelt quali laps to decide the pole, rather than probably 4 or 5 now (3 if you are in a Merc).
Plus, after each quali lap there needs to be time to sort out who is the slowest and eliminate them. Say an average lap is around 1 minute 30. You need to add 2 minutes for the out lap and 2 minutes for the warm down lap. So that is 5 minutes 30 per section. Then a couple more minutes for a splash of fuel and new boots (plus no idea how this would effect the number of tyres allowed in a weekend, as you could use 22 sets of ultra-soft) so a total time between runs of 7 minutes 30 seconds. Multiply by 21 runs. 150 minutes and 30 seconds for a qualifying session, assuming everything goes smoothly. And that assumes everyone on track at the same time for the first runs which just won't happen.
You looking at a minimum of three hours. And frankly most of the audience will only want to watch the last forty minutes.