[llvm] [libc++] Switch over to the LLVM-wide premerge test runners. (PR #147794)

[Aiden Grossman via llvm-commits]

boomanaiden154 wrote:

> As it stand now, upgrading the container is still a manual process and a bit of a pain. With work we can cleanup our Dockerfile to better support rebuilding without affecting the compiler/tool versions. But until then, this process seems manual, and we should ensure it's done with a regular cadence.

I was hoping to get to this refactoring by the end of the week.

https://github.com/llvm/llvm-project/pull/147794