It's been both parties all along. Back when Reagan signed the first amnesty, the Attorney General began to enforce the employer ID provisions seriously. A group of Republican congressmen wrote a letter to him complaining that the enforcement was hurting business and threatening to cut the DOJ budget. After that there was never any serious effort enforcing those provisions. Pretending Pubs aren't a part of the problem is the revisionist history.
2. A Democrat, Bill Clinton signed NAFA.
Most of the votes in Congress came from Pubs. Almost all the votes against were Democrats.
more Republicans voted for the Civil Rights Act too.
That’s not exactly a measure of who wanted it the most.