feat: add schema coverage demo for private (#36200)

Author: github-actions[bot]

schema-demos/schema-demo-private.md

@@ -0,0 +1,30 @@
+---
+description: Demonstrates the `private` schema field
+on:
+  workflow_dispatch:
+permissions:
+  contents: read
+engine: codex
+private: true
+timeout-minutes: 5
+---
+
+# Schema Demo: `private`
+
+This workflow was auto-generated to demonstrate usage of the `private` field in
+the gh-aw frontmatter schema. It exists solely to achieve 100% schema feature
+coverage.
+
+## What `private` Does
+
+Marks the workflow as private so it is not meant to be shared outside its repository.
+
+## Task
+
+Call `noop` -- this is a coverage-only demo workflow.
+
+**Important**: Always call the `noop` safe-output tool.
+
+```json
+{"noop": {"message": "Coverage demo for `private` -- no action needed."}}
+```