This entirely rests upon the assumption that supporting amnesty would win Republicans support among Latinos, which simply isn't the case. All the polling of Latinos shows the primary reason for their inclination towards the Democrats is economics/welfare.
But I'm not sure it's just a matter of poverty. After all Asians and Jews are both wealthier then the average white and their isn't a large body of illegal immigrants that they identify with, yet they're also overwhelming tilted to the Democrats. I think it's fundamentally a cultural thing.
Their is no scenario in which Republicans will get much headway among Latinos. They clearly won't be moved by religious conservatism, they remained Democratic inclined even after Reagan implemented amnesty, and movement to the economic left by Republicans will simply result in a parallel shift leftwards by the Democrats. Amnesty would be political suicide.
I think you might be a troll, but I will respond anyways since you made a couple of good points.
First of all you are correct that a lot of Hispanics who vote Democrat do so due to economic reasons, but do 71% of them vote Democrat due to economic reasons? What the GOP needs to do to win is get that margin down to 60% Democrat. They won't win the Hispanic vote and they shouldn't need to.
This brings me to an even more important point. The Republicans need to win the Asian vote, especially the Christian Asian vote, by double digits. They really need to do that. That will immediately make them more competitive in states like Virginia and Colorado while earning them a few congressional seats in California. Jews and Hindu/Muslim South Asians are probably gone to Democrats though, including the funding source.....but the Republicans can't do anything about that. I think the Republicans will try to transition to a christian identity party as opposed to a white identity party so these people will still be left behind. That way they can still try and keep their white working class base. Of course I am not sure how good that strategy will be in the long term as younger whites are much less religious. At least that could be a good medium term strategy.
Amnesty with immediate right to vote should never be endorsed by the Republicans. They need to be smart and give them some special immigration status which makes them residents, but doesn't allow them a chance at naturalization for 10 years. But it's not just the need for amnesty that influences the Hispanic vote, it's also the racist laws like the one passed in Arizona. The Republicans really cannot do that!!!