It's actually true that the president can send foreigners home and he can do it without any motivation whatsoever, other than because he bloody well feels like it. But it's done by specifically singling out the foreigner to be sent home, and then a presidential decree is what gets the person kicked out. The president has no time to bother to exercise this power.
This sounds markedly better than the ability to throw foreigners in Guantanamo Bay with no reason whatsoever.
PS
This sounds markedly better than the ability to throw foreigners in Guantanamo Bay with no reason whatsoever.