[clang-tools-extra] [clangd] Allow specifying what headers are always included via "" or <> (PR #67749)

[Nathan Ridge via cfe-commits]

HighCommander4 wrote:

> > It's sad that there's not a single codepath that would handle all these, but maybe the perfect is the enemy of the good here.
> 
> The code duplication was my reason for looking at the common path. If you're fine with duplicating the code I will go ahead and do that.

+1, please go ahead and wire up the include cleaner support based on Sam's outline; it seems fairly straightforward, the amount of duplication is not much, and we might as well have the config work consistently across all include-insertion features.

The new test + other changes look good as well!

https://github.com/llvm/llvm-project/pull/67749