If you combine that with looser zoning regs, the higher rents will make building more housing more lucrative.

In my mind, I was separating out homes for rent from investment and second/third/so-on homes. Although, it seems tough to separate those out since investment homes and non-primary homes are often rented out, although sometimes not permanently or continuously as homes specifically treated as rental properties. It needs some more thought, but I don't like the no holds barred approach that takes place right now. I in general don't like homes being treated as a business and investments, and I think it should be discouraged. Apartment complexes are a little different. Even there though, renters need better protection. One of my friend's rent just increased by 25%. It's ridiculous.

Make the tax scale with number of homes the entity owns.

They might not be able to find someone to pay that rent. In which case they might be forced to sell. Ideally someone who doesn't own a home will purchase it and won't have to pay the tax that caused the original owner to sell.