That's missing in this entire conversation. This all sounds like it's 2016 or something
At some point, we have to accept that the world has irrevocably changed, that people have lived where they live for generations, that resettling them all is impractical, inhumane and just not a good idea in general, and just move on and build the best world for the people currently here.
During WW1 and then again during WW2, the world has agreed to the right of self-determination. If people feel they're a certain nationality separate from what their government imposes on them, then they have a right to become their own country with their own laws and their own border enforcement. If you are literally asking what gives Americans the right to prevent others from coming, this is it.