The GOP is doomed IMHO. They’re just like the old Whig party from before the US Civil War; they have no clear stance on any moral or economic issues, they’re internally divided, and they’re obsessed with fighting foreign wars to maintain and expand the empire while the domestic situation is deteriorating.
I don’t think Rand Paul would change that much, even if he were to get his way in the party. He’s a libertarian, which means he would be on the far left socially and the far right economically. That means gay “marriage”, abortion on demand, very low minimum wage (if any at all), and a lot of other things that are incompatible with either Catholic moral theology or Catholic social teaching.
One thing that could fix the GOP is if they were to become like European conservatives, like the traditional Christian democrats–that is to say, on the center-right socially and the center-left economically. Such positions would probably appeal to Catholic and Evangelical/Pentecostal Christians of color, who are a growing political voice in the US, as has been implicitly pointed out by Geist.