It comes down to which wave they came in on.
Reagan was an inspiration. Immigrants can be inspired by a great leader of their new country just as much as the native people. Those that came in under Reagan are Reagan Latinos. And they do vote Republican.
The next wave came in under Clinton/Bush. What was their inspiration? New Latinos have been cultured into a very different world -- an uninspiring world.
That same thing has happened to millennials. Worse, they're cynical. They don't start families. They're unemployed.
If we can find someone that can inspire millennials, we'll have no trouble inspiring new immigrants. Amnesty wouldn't be a huge problem with such a leader.
But where is that leader today?
Amnesty is not the main problem. Our legal immigration policies are far more destructive.