I do think we still need to build more (correctly sized) housing, counteract urbanization, provide social housing, and I'm in favor of rent control in general. So don't get me wrong, I don't think progressive tax will solve the problem on its own. That being said:
Higher taxes could lead to higher rents, but that stops working once the supply of people willing to pay said higher rent is exhausted. At some point landlords won't be able to rent out their place with a profit, at which point you have 2 options: Leave it empty (which should also be illegal, IMO) or sell it (to people who would otherwise have to rent).
It could be that there is a potential never-ending supply of rich people wanting to pay rent to live in a place with ever-increasing rent and the gentrification that comes with it, but it seems unlikely to me.
In the Netherlands there were recently a set of tax changes increasing taxes on house ownership, which caused companies to sell apartments en mass. Housing prices dipped for a little bit, rent (in the private market) went up, but more people own their own place now. This is the direction you want to go, I think.