ci(workflows): fix registry sync and link check

[kantorcodes]

Summary

• replace the dead PR badge target with the local Contributing section so markdown link check is stable
• remove the invalid secret-context conditional from the cross-repo sync workflow
• add workflow_dispatch so admins can manually smoke-test the sync workflow after it lands

Root cause

• Build & Deploy failed because README linked to http://makeapullrequest.com, which markdown-link-check reports as dead
• Sync to awesome-ai-plugins failed before jobs ran because GitHub Actions does not allow secrets context directly in a step if expression

Verification

• yq parsed .github/workflows/sync-to-ai-plugins.yml and .github/workflows/deploy.yml
• npx markdown-link-check@latest README.md --config .mlc_config.json
• node build.js

[gemini-code-assist]

Code Review

This pull request updates the 'PRs Welcome' badge link in the README.md file to point to the internal contributing section instead of an external URL. I have no feedback to provide as there were no review comments.