The house analogy makes zero sense, as my house is not a commons (I am sole owner, even with a mortgage, methinks you don’t have one of these), but if it were then yes, anyone agreeing to the social contract of the government of my home would be allowed in or out.

It’s about how to manage shared resources amongst people who all have equal claim, which is everyone who has agreed to the social contract. When you agreed is irrelevant.

Frankly, I would argue that many people born in the US do not hold up their end of the social contract, and should thus take a backseat to immigrants who would come here and care deeply for the social contact in their place.