Housing and land require further analysis due to inelastic supply of land, and huge restrictions on construction, both for political and safety reasons.
The biggest determinant of supply of housing is not the ability to charge more for it, but all the other challenges around building.
So it's not a "price control" as much as it is a control on rentierism, on existing rentiers taking unearned profits without working.
Here is an idea: Measure average rent per sqft. Put a 20% tax on rent above the average and use it to subsidize low income households. That way you tax wealthy tenants and subsidize poor tenants.
What there is is a lack of accommodation where people want to live.
Ultimately people need to be persuaded or given a good reason to live elsewhere.