Interesting and provocative post, Jo. Unquestionably, the GOP is as scattered on immigration as Romney was on nearly every issue. How to fix that? I don't know, but at its' present state, our national immigration policy is in shambles.
But that's just it, it's not all immigration.
Who is the most actively revered figure in Latino Catholicism? It's not Jesus, it's Mary. The most successful Protestant missionary movements in Central and South America have been ones that took on a Marian theology. These Hispanics aren't as taken by Italian-style guilt and threats of hell for sinning or Polish-style stoicism, they're attracted to Marian ideals of love, compassion, helping the downtrodden, caring for the weak, etc. And that attitude isn't purely a religious one but one that has become cultural.
Republican conservatism doesn't fit. Conservative ideals of "You should be charitable but the government shouldn't be" is like being told "Well, you should give but the Church as an institution isn't going to give anything to the poor. You do that yourself." The Republican stance on immigration (such as there is one) is hostile to those ideals. It splits families and attacks the poorest and most vulnerable. Attacks on welfare, social services, unemployment, etc are much the same.
That's absolutely not to say that they are not hard working or want
to be on those programs. But rather their philosophy is that the programs should be there and should be strong. Democrats support that idea more than Republicans do.
I find it interesting because I've always subscribed to the same myth the GOP does, that if you just smoothed out immigration, Hispanics would culturally become good socially conservative Republicans. Catholicism was always the keystone of that argument. But I was looking at it through the lens of my own Polish Catholic "Keep quiet and get back to work; you're a good Catholic by working hard and following the rules" upbringing. And I think most Republicans expect all the other Papists to follow the same sort of European rules (along with the Italian guilt theology) and I don't think it holds.
I'll footnote by saying I'm glossing over very Democratic aspects of European Catholicism as well since they are somewhat irrelevant to my greater point.