There are also large groups of people- whites and Hispanic both, that think all Hispanics here came here illegally. Ignoring the History that many are descendants of Hispanics that were here before this was the United States, and many others that have legally immigrated. The other day a person with a Hispanic surname was posting on facebook that they were against any plan that rewarded illegals for breaking the law. Several other Hispanics tried to slam the person by saying since they were Hispanic at some point their family was also illegal- which of course may not be true.
So many myths out there- Republicans hate Hispanics, all Hispanics are illegals, there is no way for Hispanics to legally come here to work or live or become citizens. All of which shows how effective the supporters of illegals/Democrats have been at changing the language and getting the myths out in common belief.
I think you are right -- I also think that immigration is the key issue, but not in the way that most people think.
It's not that most Latinos support illegal immigration -- they don't -- it's that a lot of conservatives use racial language and cultural slurs when we oppose it and that's obviously deeply offensive to Latino US citizens regardless of how they got their citizenship.
Heck, even Cubans, none of whom are illegal and a population that was once the most Republican group in America are now trending Democrat, and I think the evidence is this is because of the perception of anti-Latino racism on the part of the GOP.