I respect that you want a temporary solution and are putting in effort, but what you suggest would just make the problem worse, it would mean that a much larger percent of people are playing in public lobbys, ESPECIALLY new players since they are the ones who aim to get Qi, Then one high skill player would enter and streak them all. Me and you Powas have very little reason to play for Qi now, since nothing really comes from it, but to a blue belt it can mean a lot. Yes you are correct when you say the majority comes from QA rooms, i had forgotten they existed, they are the heaven of farming.
The idea of people setting private lobbys and farming rank and QI in jousting is kinda old fashioned now and not that common.

Mwah I originally suggested a matchmaking suggestion where people could veto mods but i quickly realised that would be far to much work for Nabi, The majority of your idea seems very similiar to the one I suggested, but I just made it easier to be implemented. A matchmaking system with being able to Veto a selection of mods would be far greater to my idea but I cannot see it happening.

If you look at the parts that it takes to make the system I suggestion possible there are very few 'new' ideas. The only one being an automatic mod rotation. (Which is already possible with bots) and Rank restriction from lobbys, which is already very similar to QI restriction and probably just needs a few tweaks in the code.