I believe the shift arrived when President Reagan gave clemency to 3.5 million illegal immigrants and the reaction from the extremes of conservatism has gained traction ever since.

Kinda. Reagan had a path to citizen ship, but in exchange for stopping a free flow in and out of workers. Once they made it, in they had to stay full time instead of come for seasonal work. This has lead to them being a full underclass on the run from the law, and all the fearmongering with that.