I've thought about the avg lap times being your qualifier, but I don't think they do that, in reality, do they? (If so, someone educate me, please. )
BUT, I do think, you should have to do a qualifying lap for every race. I don't think you should get to run a qualifier at 8 AM and then be able to do 20 races all day with that one qualifying lap.
Correct, they do not do this in real life, but, in real life qualifying takes hours, and you make many laps, and you cannot wreck you car in the name of ultimate lap. They also have several practice sessions, with many more hours of running on track.
My opinion is that if you at least forced people to put together 5 consecutive clean laps to enter a race you are doing a few things;
forcing them to be come familiar with the track/car combo.
forcing them to do so at a repeatable(race?) pace.
forcing them to not simply smash/crash their way to some magic lap where it somehow all came together but is in no way repeatable.
In doing the above you are starting a trend of muscle memory... which... I don't know, just might make better racing and matchmaking.
As for penalties being harsher. Not sure the penalties will ever suit or make everyone happy. Want to clean up the game? Want to get rid of the idiots who like to dive bomb and take you out of the race, while they can just continue on and deal with a 5 sec penalty? Turn mechanical damage on. Period.
The best/cleanest racing we've seen was under penalty system in P1.13, which was very strict, and did not try and place blame/innocence.
All cars involved got penalized, no way to "game" that.
For a few days the racing got extremely clean and gentlemanly.
Unfortunately, the player base is mostly folks who want to dive/barge/bang/crash their way to the front as opposed to develop observation skills and race-craft that goes with it, so, they all screamed for their lives and we've had nothing but gamable patch after gameable patch. again.
the above are again, my opinions, all are free to have their own.