docs: add router QA dry-run example

[saurabhhhcodes]

Summary

• add Example 9 to examples documentation and update the example count
• add a runnable router QA dry-run script that validates fallback decisions without real API calls
• support both installed-package usage and local checkout execution after build

Fixes #35

Validation

• npm run build
• node examples/9-router-qa-dry-run.js
• git diff --check