> it has a long history of suspicion of national government IDs.
I think it's more that the people in power and their supporters have had reason to make government IDs hard to come by for the transient, poor, and other undesirables. Or they simply don't want to 'muh tax dollars' going to the 'lazy bum' on the street getting a government ID card. I think there's also been a fair amount of pushing by the auto industry to make drivers licenses the default ID, too - starting in the fifties you really saw the "you have a car if you're important/Made It" attitude creep into US society.
By the way: did you notice that the politicians shouting loudest about illegal immigration tend to be associated with industries and states that use lots of illegal immigration (like, saaaaay, farming?)
It's not about stopping illegal immigration. It's about them having power over their workers. "Illegals" don't talk to police, public health officials, DAs, OSHA, etc about pesky things like missing/deducted paychecks, workplace safety, violence/theft, etc.
"They're stealing your jobs" is just how they sell it to Joe Q. Public, because one way to win someone over is to tell them that there's someone further down the ladder than them.