There's a simple solution to the overcrowding...an impenetrable wall/fence on the border...and the whole problem disappears.
We went through this once before, Bambi-ignorance, and you said you got it, so let's try it again.
Let us suppose your impenetrable wall and let us likewise assume that while it cannot keep out either the majority of drugs or the majority of folks living here without appropriate documentation, that it can keep out those from the south.
This still leaves thousands and thousands of homeless, terrified people searching for safe haven. It is not in Mexico. It is not in Honduras. It is not in Nicaragua. It is not anywhere they could get to.
You would have them die, uncared for and unsupported.
Or, as you put it, "the whole problem disappears."
Simple.