Maybe each team should have an alloted time to run 3 single (or double or whatever fiasco tyres happen to create) fast laps then the teams with the combined fastest laps (or maybe just the drivers with the fastest) are given a final quali section to get pole.
Guaranteed cars on track, guaranteed trying really hard and a fun end!

Thoughts??

I've edited this to add to the first part:

Teams of a similar pace are alloted times to run together so say 6 cars at a time and limited time to run to force cars out for each full session.