Remove Egress Gate from public research repo

[johnnygreco]

Summary

• remove the Egress Gate project and its dedicated CI workflow
• remove its published documentation, navigation, staging, and project-index integration
• update remaining documentation tests and examples so the public repo is self-consistent
• transfer ownership to the internal research repository

Audit

• three independent review lenses completed with no remaining findings
• binary-aware full-worktree scan found no remaining Egress Gate paths or content outside Git history
• strict documentation build completed with no issues
• generated Egress Gate documentation route is absent

Validation

• scripts/build-docs.sh
• focused documentation tests: 27 passed, 4 skipped before rendered-build checks
• git diff --check origin/main..HEAD

[github-actions]

Documentation preview

The preview has been removed.