devops: authenticate stable-test-runner npm registry in ESRP pipeline (#42146)

Author: yury-s

.azure-pipelines/publish.yml

@@ -59,12 +59,18 @@ extends:
             targetType: "inline"
             script: |
               echo "registry=https://devdiv.pkgs.visualstudio.com/DevDiv/_packaging/DevDiv_PublicPackages/npm/registry/" >> .npmrc
+              echo "registry=https://devdiv.pkgs.visualstudio.com/DevDiv/_packaging/DevDiv_PublicPackages/npm/registry/" >> tests/playwright-test/stable-test-runner/.npmrc
 
         - task: npmAuthenticate@0
           displayName: "authenticate the private npm registry"
           inputs:
             workingFile: .npmrc
 
+        - task: npmAuthenticate@0
+          displayName: "authenticate the private npm registry for stable-test-runner"
+          inputs:
+            workingFile: tests/playwright-test/stable-test-runner/.npmrc
+
         - script: npm ci
           displayName: "npm ci"