What's wrong with waiting lists? Sounds much fairer than to give all housing to rich folks. I'm not advocating that 100% of housing should be run this way, but for sure more than what is now (in most large EU cities).
I am saying "The problem is not that there isn't sufficient housing" is wrong. Very very very wrong. There are massive supply shortages of housing in many many places. Governments need to fix their oppressive zoning policies so they stop preventing people from living where they want to live (either because of $, because of waiting lists, or any other mechanism).