[libc++] Rewrite the transitive header checking machinery (#110554)

Since we don't generate a full dependency graph of headers, we can
greatly simplify the script that parses the result of --trace-includes.

At the same time, we also unify the mechanism for detecting whether a
header is a public/C compat/internal/etc header with the existing
mechanism in header_information.py.

As a drive-by this fixes the headers_in_modulemap.sh.py test which had
been disabled by mistake because it used its own way of determining
the list of libc++ headers. By consistently using header_information.py
to get that information, problems like this shouldn't happen anymore.

This should also unblock #110303, which was blocked because of
a brittle implementation of the transitive includes check which broke
when the repository was cloned at a path like /path/__something/more.

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