Good points.

All variants of rent control etc. have been tried in Europe and have miserably failed. Quite the opposite, rents have been rising even more, and new construction has been reduced due to new politically induced risks.

Examples: Berlin, Barcelona

But as Barcelona shows, there is a feedback loop benefiting leftist populist politicians:

Higher rents, lower housing supply -> people frustrated -> leftist populists get more votes -> more stupid regulation -> even higher rents, even lower housing supply -> people more frustrated -> ...

This can go on for at least two electoral cycles.