Born within US borders, what gives me (and you) the right to turn away people born outside US borders?

You almost certainly didn’t earn your citizenship. Even if you’ve served in the military or the police, you earned your citizenship as an adult; you got a lucky break, being born here, through no virtue of your own. Where do we get the idea that we can tell people they’re not allowed to come here?

Whatever mechanism it is that gives us that right, why didn’t it give the North American aboriginals the right to keep our forebears out? How is this not a grotesque double-standard?