CI: remove separate check-build-tools-image workflow (#9708)

## Problem

We call `check-build-tools-image` twice for each workflow whenever we
use it, along with `build-build-tools-image`, once as a workflow itself,
and the second time from `build-build-tools-image`. This is not
necessary.

## Summary of changes
- Inline `check-build-tools-image` into `build-build-tools-image`
- Remove separate `check-build-tools-image` workflow

.github/workflows/neon_extra_builds.yml

@@ -26,15 +26,9 @@ jobs:
     with:
       github-event-name: ${{ github.event_name}}
 
-  check-build-tools-image:
-    needs: [ check-permissions ]
-    uses: ./.github/workflows/check-build-tools-image.yml
-
   build-build-tools-image:
-    needs: [ check-build-tools-image ]
+    needs: [ check-permissions ]
     uses: ./.github/workflows/build-build-tools-image.yml
-    with:
-      image-tag: ${{ needs.check-build-tools-image.outputs.image-tag }}
     secrets: inherit
 
   check-macos-build: