And your property rights come from the contract you have with the state. It's all in the formal and informal contract that govern society, and you can(as most suburbanites have) sign away all of the "control" that you seem to argue property gives you to some collective body.
In the same vain the government(and some wery much do) can set pretty strict rules on what restrictions a private landlord can put in rental agreements and that's before you remember that the government itself have historically been the largest owner of rental properties.
In the old days before the idea that owning a house was a ticket into a higher strata in the class system a lot of the problems now caused by unreasonable housing prices was solved by the government acting as the reasonable landlord, essentially curtailing the amount of shenanigans some wannabe aristocrat could get away with before going bankrupt from people not putting up with the abuse.
In a pure "realty don't matter" libertarian mindset your of cause right that property rights are always supreme but in the real world it's always a balance of power and negotiations especially once we deal with urban communities(which is the only ones where property prices are a problem).
Rights of property ownership is always a superset of the rights of renting property. It is not a false argument.