[engsys] inline eslint step into analyze template (Azure#32304)

### Packages impacted by this PR

None directly, this only touches CI.

### Describe the problem that is addressed by this PR

We were seeing odd behavior in CI where the `rush install` inside of the
lint step was failing, but since this task was only ever called from
analyze, which has already executed `rush install` it seemed
unnecessary.

This change inlines the behavior from the run-estlint step and avoids
the duplicate `rush install` call.

eng/pipelines/templates/steps/analyze.yml

@@ -68,9 +68,10 @@ steps:
       node eng/tools/rush-runner/index.js build $(ChangedServices) -packages "$(ArtifactPackageNames)" --verbose -p max
     displayName: "Build libraries"
 
-  - template: /eng/pipelines/templates/steps/run-eslint.yml
-    parameters:
-      ServiceDirectories: $(ChangedServices)
+  - pwsh: |
+      node common/scripts/install-run-rush.js build -t @azure/eslint-plugin-azure-sdk -t @azure/monitor-opentelemetry-exporter
+      node eng/tools/rush-runner/index.js lint "$(ChangedServices)" -p max
+    displayName: "Build ESLint Plugin and Lint Libraries"
 
   - pwsh: |
       node eng/tools/rush-runner/index.js check-format $(ChangedServices) -packages "$(ArtifactPackageNames)" --verbose