My house and the country are not analogous concepts though. I own my house, I do not have ownership over the country

In this case, they literally are analogous concepts. You don’t own the entirety of the US, true, that’s why it’s an analogy. You own your house the same way the collective of American citizens “own” the US. And our representative politicians (theoretically) make the decisions based on the wants and needs of the “owners” (citizens).

Do you really think that ppl will illegally come to the US and stay for 6 months specifically for the healthcare?

Absofreakinlutely they would. You think healthcare in places like south and Central American countries is as good as what the US offers? Especially if they can get it in the US for free?