If the Reps would address the border issue sensibly, they might win a majority of the Latino vote, and that would be devastating for the Dems.

I'm not sure that their outreach efforts depend all that much on having a "sensible" border policy. At least, not any more. Another consequence of the migrant population shifting from Mexican workers to Northern Triangle asylees is that the demographics of border control have shifted. The overwhelming majority of Latinos in the U.S. are Mexican or Puerto Rican. The latter aren't directly impacted by border issues, and the former aren't going to be as invested in the issue when it's mostly affecting different Latino communities, not their own.

The GOP has been doing an amazing job with grass roots outreach to Latino communities in Florida. Not just among the Cuban exilios down here in Miami, but all the different groups that are growing in Central Florida. Trump took 46% of the Latino vote in Florida in 2020 - I think he'll get a majority in 2024. They're not doing it with changes on federal policy issues - they're doing it with increased outreach and resources to Latino voters at the very local level, using their stronger position in state and local office to encourage Latino voters to participate in the GOP.