Higher rent means more people will opt to buy. That's not necessarily a bad thing in perspective. Assuming the other side effect is a decrease in costs due to a lower profit margin.

That is a good point, and this tax would need to be well thought out. I specifically would increase the tax rate the more properties an entity owns (Not a lawyer so IDK how to treat one guy owning 10 companies as 1 entity the right way). That way smaller local landlord's can still exist (and keep rental prices competitive), but massive orgs will be forced to sell properties. The idea is to create a disincentive for hoarding properties in supply constraint markets.

Long term these non-occupants will sell as other properties are being purchased by x-renters.