Your post raises and interesting question:
We [USA] recognize the right of people to the pursuit of life liberty and happiness [property]. It doesn't say that Americans rights are recongnized...it says the peoples right is recognized.
I suppose some crafty liberal lawyer could twist this into a right of trial...
The point is that the protections being cited as a basis for claiming that illegals have the right to a trial before deportation is the 14th amendment. The 14th prohibits any
state from depriving any person within its jurisdiction of life, liberty or property without due process of law or to deny any person therein equal protection under the law. It would appear that the lawyers would rather we didn't realize that those provisions of the 14th are aimed only at the state governments. No person who stands to be deported has life, liberty or property put at risk by the action of the US in deporting them. They may pursue happiness all they like as long as they don't violate the laws of whatever jurisdiction they may choose for their pursuit.
So, the statutes which require that an alien get a trial to determine whether or not he is to be deported is all that even puts an alien on trial where the federal government must treat his privilege of visiting as a right. Repealing the statutes completely solves the problem because the Constitution doesn't compel government to recognize the right of foreign nationals to invade the US.