The famed economist Milton Friedman observed years ago that you can have a welfare state or you can have open borders, but you can't have both.
In the real world, we're not dismantling our welfare state. Too many people are dependent up on it and many people, like those approaching retirement, have legitimate expectations of getting something back for decades of taxes paid into it.
The idea that all illegals, or even most of them, are hardworking model citizens is a fantasy. Many are on welfare of one form or another, many are involved in gangs and crime and they have a disturbing tendency to kill legitimate Americans, wither through crime or drunk driving.
A central aspect of sovereignty is the ability to choose who enters your country. It is perfectly legitimate and reasonable to say we don't need or want large numbers of uneducated, low skill people who do not speak our common language and who do not share our culture.