Some very good points made above.
I would like to see one of the following adopted, though I know they never will.
1 hr qualifying, free for all.
Cars can be used by either driver.
The hour is divided into 3 continuous 20 minute segments.
In the first segment, failure to complete a lap within 107% of the fastest lap set within that segment will eliminate you from the meeting. - Remember, at Bahrain this would have been around 8 seconds off the pace. If you can't do that, you really shouldn't be there. Similarly, Interlagos will see you about 6 seconds behind.
In any segment, failure to complete a lap within 105% of the fastest time within that segment will eliminate you from the next segment.
It may require the end of the engine rule, or altering it to be 16 engines per team per season if they have to keep a limit, but other than that it requires all the top cars to be out in all sessions, whilst removing the mobile chicanes, with a 107% rule that is relevant to the current conditions.
Further, being able to borrow your teammate's car gives you a lifeline in case you stuff it in the wall/have a failure.