[clang-tools-extra] [clang-tidy] Fix handling --driver-mode= (PR #66553)

[Julian Schmidt via cfe-commits]

https://github.com/5chmidti commented:

I don't know enough about the driver to approve this, but I read this a few times and I think that this is correct. I just added a few small and non-functional nits

https://github.com/llvm/llvm-project/pull/66553