"Tell me how price controls would cause less people to be housed"
Bourgeois economics will tell you that setting a price ceiling below market value means it's less worthwhile to build new housing so there's less investment in it, which means demand doesn't keep up with supply.
Now, the issue with this in the reality we live in here in the USA, at least, is that, if given free license, real estate developers will build high-end luxury condos because they're the most profitable. YIMBYists will tell you this is fine because the wealthiest people will move into those condos, and less wealthy people will move into the newly vacated housing. Everyone "inherits" the housing a rung up, and everyone's living conditions are a little better. In reality, this process takes ages, and since demand is so high, it means that the new stock is snapped up by wealthy out-of-towners anyway, and a lot of the middle and lower tier is just straight out bought up by investment firms who would rent it out and suck up all the real estate capital for themselves.
This can be circumvented by public ownership. Here's a good write-up of Vienna's public housing situation. By owning the bulk of the housing and leaving it open to residents at a low cost, you can force developers to either match your prices, or cater to other niches like the wealthy, which, in a bourgeois economy, is honestly fine.
