Try using clangd-tidy (#5763)

Run clangd-tidy in parallel with clang-tidy, to experimentally see
whether it works reasonably well. These may produce slightly different
results, and it's not clear that clangd-tidy will be better, so being
cautious about switching.

A real possibility here is this is slower in some cases (building
compile commands takes ~6m below), but faster in the extremely slow
cases (when clang-tidy takes >10m).

For contrast:

- clang-tidy:
https://github.com/carbon-language/carbon-lang/actions/runs/16038096026/job/45254162180?pr=5763
- clangd-tidy:
https://github.com/carbon-language/carbon-lang/actions/runs/16038096427/job/45254164217?pr=5763

.github/actions/test-setup/action.yml

@@ -19,6 +19,8 @@ inputs:
 outputs:
   has_code:
     value: ${{ steps.filter.outputs.has_code}}
+  has_cpp_files:
+    value: ${{ steps.filter.outputs.has_cpp_files}}
 
 runs:
   using: composite
@@ -32,6 +34,8 @@ runs:
         filters: |
           has_code:
             - '!{**/*.md,LICENSE,CODEOWNERS,.git*}'
+          has_cpp_files:
+            - '{**/*.cpp,**/*.h}'
 
     # Disable uploads when the remote cache is read-only.
     - name: Set up remote cache access (read-only)