I know the nazi germany trope is so played out it's become a joke, but the nazis also were looking at "deporting" undesirables until they realized that was impossible, and after that they started opening the concentration camps (though whether the camps were always supposed to be the end goal or not is debatable). Like you said the country of immigration decides who to let in, and it will be up to them if they accept non-citizens, and with or without visas. But the US can't tell Mexico to start taking in everyone who crosses the border, Mexico chooses that.

Consider also that ICE is already at the border, and you already have to present a passport to board a plane. There is no reason they can't repurpose this system (or make an entire new program, thanks to DGE currently having all the powers it wants to have) to start enforcing travel bans outside the country. I just remembered right now as I was typing that they already prevent travel to countries like Cuba or the DPRK.

I wish you all the best though. It's a terrible situation, and it's coming to Europe too (funded by the US in fact).