[release/5.0] Test in manual internal builds (#36484)

- backport of #36424 to release/5.0
- make it a bit easier to test _everything_ in one go
- builds site extensions, publishes to BAR, and runs all aspnetcore-ci tests

Co-authored-by: Doug Bunting <[email protected]>

Author: github-actions[bot]

.azure/pipelines/ci.yml

@@ -117,7 +117,7 @@ stages:
   displayName: Build
   jobs:
   # Code check
-  - ${{ if or(eq(variables['System.TeamProject'], 'public'), in(variables['Build.Reason'], 'PullRequest')) }}:
+  - ${{ if or(eq(variables['System.TeamProject'], 'public'), in(variables['Build.Reason'], 'PullRequest'), in(variables['Build.Reason'], 'Manual')) }}:
     - template: jobs/default-build.yml
       parameters:
         jobName: Code_check
@@ -595,7 +595,7 @@ stages:
       parameters:
         inputName: Linux_musl_arm64
 
-  - ${{ if or(eq(variables['System.TeamProject'], 'public'), in(variables['Build.Reason'], 'PullRequest')) }}:
+  - ${{ if or(eq(variables['System.TeamProject'], 'public'), in(variables['Build.Reason'], 'PullRequest'), in(variables['Build.Reason'], 'Manual')) }}:
     # Test jobs
     - template: jobs/default-build.yml
       parameters: