Really good question and I've wondered the same thing. Specifically, why isn't E-verify mandatory for all employers of any size. It was a key 2016 campaign promise.
This is how enforcement of immigration laws is done in other countries, e.g. the UK and European Union. You'll still get illegal immigrants hanging around on street corners looking for odd jobs to do for cash under the table, but that won't be enough for them to live here on a long-term basis, let alone raise a family here.
Ending welfare might help, but it won't be enough by itself. Illegals will still show up in ERs, and blue cities will continue to give them benefits like food stamps and housing. And the major draw for them will still be the ability to get high paying jobs (compared to what they could get in their home countries).