Some are for sure but lots work with stolen SSNs or TINs. Most businesses want at least plausible deniability when ICE visits. Saying they all work under table is as silly as saying none do.
But this as nothing to do with who the dems should run for the presidency. I think it should be a moderate governor of a medium sized purple or light red state.
Sure, Trump was scare mongering with illegal immigrants and trans atheletes but the Dems have staked out a lane on those issues that is favored by 20 to 30% of the electorate at most. With trans athletes, the 80% that think its unfair for biological males to win triple jump by 8 feet think you are crazy when you say they don't have any unfair advantage over biological females. It not just unpopular its a position that most people think indicates the dems are basically delusional on. Then you take say, Newsom, and he signed a bill that made it illegal for a teacher to tell a kid's parents that they've transitioned at school* and think he'd be an electable candidate outside of CA or NY?
*Actually, I've read a few articles on this, its unclear if it prevents schools and teachers from revealing gender transitions to parents or just bans school board from mandating that schools reveal gender transitions. Point still stands, the GOP can sell it as the first regardless, and most folks will think its crazy either way.