[clang][cli] Parse HeaderSearch options separately

This patch moves parsing of header search options from `parseSimpleArgs` back to `ParseHeaderSearchArgs` where they originally were.

The round-trip algorithm in D94472 requires this separation to be able to run parsing and generating **only** for the options that need to be tested via round-tripping.

Reviewed By: dexonsmith

Differential Revision: https://reviews.llvm.org/D94802