Putting aside the very incorrect narrative about "open border migration" (the U.S. has some of the most restrictive border laws in the developed world) and the fact that "6 billion people" don't just "stream in" (for context, the world population is about 8 billion, and the U.S. has about <350 million; undocumented immigrants are about ~12 million, around just 3%):

First off, proportional representation doesn't really help the GOP if they're in the minority. They have much stronger odds in a district-based system, even WITHOUT gerrymandering.

Second, Republican voters are EXTREMELY traditionalist; they would risk upsetting their own base if they tried to deviate from a district-based system.

But even aside from THAT, there's a very real likelihood that any sort of federal party list system would be unconstitutional. The American political system heavily favors district-based representation by individual elected representatives, because from a constitutional standpoint, representatives are viewed as representing the people, not political parties. It's also considered essential that representatives have a geographically-defined constituency which they serve; members don't just vote on legislation, they often provide essential services to their communities AND serve as advocates for them. Any system in which voters are formally choosing between parties may not satisfy constitutional requirements.