If you instantly removed all undocumented aliens from the USA our farming sector would collapse. To put it bluntly, citizens will not do the work they do for the wages they get.
Which is a very strong argument for immigration reform around migrant workers.
he vast majority of them are gainfully, if illegally, employed, paying taxes and social security through phony green cards and SSNs.
That's not true. The vast majority are taking jobs where they get paid under the table in cash and zero taxes are being paid. That's not actually a problem as the pay is normally too low to really bother with taxes.
The two issues are the fact that very few states check for immigration status before allowing welfare. That family of illegals where the dad is working an illegal construction job, they are receiving WIC and other financial assistance. The second issue and by far the worst one, the male geographic bachelors working those jobs living six to ten in a cheap dwelling sending the money "back home". That is extremely bad, stupidly bad for the economy. Currency is the life blood of a nations economy, it must circulate continuously. People sending money "back home" are effectively leaching economic value out of that nation and redistributing it to a poorer nation. One leech can't hurt you, hundreds or thousands of leech's will kill a human.
Which is why I'm very much for an overhaul of our immigration system to make it faster and easier for visa's to be processed and legal status to be obtained or denied. Having gone through the system myself when I sponsored my wife, I spent thousands of dollars, took almost a year and we were in the super fast "immediate relative" expedited lane. Others doing it legally takes years and years to process. Our lawyer told us she had one client that's case as been working it's way through the system since around 2009.
So while I'm extremely anti-illegal immigrant, because ya know I had to go through the whole thing the hard way, I'm also very much about making things faster and getting some sort of legal status for those who are here. Not citizenship mind you, but a form of legal status that tracks their presence and can enable them to legally contribute to our great nation.