[libc++][CI] Tests the no RTTI configuration. (#65518)

There are a few drive-by fixes:
- Since the combination RTTI disabled and exceptions enabled do not
work, this combination is prohibited.
- A small NFC in any fixing clang-tidy.

The code in the Buildkite configuration is prepared for using the std
module. There are more fixes needed for that configuration which will be
done in a separate commit.