You ignored the crux of my point.
If you do not want undocumented workers sneaking and staying in illegally and working then the easy solution is to punish the corporations who hire them. Full stop.
Not only does it not cost the tax payers anything but it makes them money if you do it that.
You tell Trump Co's in construction, and Hotels that if they continue to hire these people every time the spot light goes off them only to keep enjoying higher profits due to paying them lower wages, that the gov't is not only going to take back those increased profits, via fines, but multiple offenders will lose their business charter and be shut down.
That is an easy fix and one that costs the citizens nothing and that would be near 100% effective.
So whenever you see a gov't not take an easy and cost effective or profit making initiative and instead suggest something that will not work but will require the gov't to spend hundreds of billions, you should realize why they do it. Don't be a dupe for gov't. The gov't does not want to take away the profits, due to low cost labour from employers. They want to KEEP the undocumented workers around as they are a great scapegoat for other problems, and they want to spend billions on useless solutions padding the wallets of donors and friends.
People need to wake up. The answer to this is painfully easy. Fine and shut down repeat offender employers and this problem goes away instantly. Full stop.