[clang] [llvm] [clang][ci] Move libc++ testing into the main PR pipeline (PR #93318)

[Vlad Serebrennikov via cfe-commits]

Endilll wrote:

https://buildkite.com/llvm-project/github-pull-requests/builds/67044#018fab83-257d-446a-9ec2-3faab776dfa8 has both new additions and `clang-ci` run at the same time.
`clang-ci` took 15 minutes, Linux run of GitHub Pull requests pipeline that does what `clang-ci` does on top of what it has been doing took 22.5 minutes. It used to take anywhere between 8 and 12 minutes based on a recently completed builds I see.

Based on the steps that we don't need to do anymore, we should be saving 4-6 minutes per build:
1) Checking out the repo to generate the pipeline (2 minutes)
2) Build Clang (3.5 minutes)
3) Upload the artifacts, download them 3 times and unpack 3 times (0.5 minutes)

https://github.com/llvm/llvm-project/pull/93318