Agreed, I think the simple answer is the tax rate is one thing for a primary residence and another for non-primary residences regardless of who owns it. For example in CA, Prop 13 stays in place for your primary residence, but properties are re-assessed every year like in Texas if it's not your primary residence. In addition, take away some (or all) of the tax deductions for SFH that aren't primary residences.

Rent would go through the roof and it would become even harder to get out of the rental trap. It's better to just have a very low hard cap on the number of properties anyone can own.

People being less likely to buy multiple houses is going to make it harder for people to buy a primary residence?

You think landlords won't pass their costs on to tenants? How do the tenants save when their rent is even more unaffordable than it is now?