docs(gates): widen relative-link checker to all docs, repair the rot it finds

TestDocsRelativeLinks walked only docs/developer; widening it to all of
docs/ immediately surfaced the same class of rot the gate was built
for, outside its old reach:

- docs/index.md was the old monolith's landing page: linked the deleted
  api-reference.md, 10 directory-style area links, and escaped the docs
  tree to README/AGENTS/.env/compose. Rewritten as a portal whose every
  link resolves, with one representative endpoint doc per API area and
  pointers to the new writing-ui-docs.md standard.
- directory-style links in user stubs (devices/, ../operations/)
- case-mismatched links (Development.md) in contributing.md
- monolith-era pointer (API_REFERENCE.md, TROUBLESHOOTING.md) and
  escaped AGENTS.md links in contributing/development.md
- docs/development/calibre-opf-implementation.md double-up escapes, a
  link to the removed repo-root plan file (absorbed here), and a
  source-code hyperlink the docs site can't serve (now inline code)
- operations/operations.md linked three never-written guide pages as
  aspirational links - they stay plain text until pages exist

HANDS-OFF (project owner, 2026-11-02): docs/user/filters/** and
docs/user/searching-and-filtering.md are exempt as link sources so no
future checker failure can force edits to the filter documentation.
Files linking INTO them remain checked.

docs/contributing/development.md

@@ -510,9 +510,9 @@ DEBUG=true
 
 ## 📚 Additional Resources
 
-- [AGENTS.md](../../AGENTS.md) - Agent/development rules
-- [API Reference](../API_REFERENCE.md) - Complete API documentation
-- [Troubleshooting](../TROUBLESHOOTING.md) - Deployment issues
+- [Development rules and standards](contributing.md) - Contributing portal
+- [API Reference](../developer/api/api-reference.md) - Split endpoint docs, indexed from its overview
+- [Troubleshooting](../operations/troubleshooting.md) - Deployment issues
 - [Go Documentation](https://go.dev/doc/)
 - [Echo Framework](https://echo.labstack.com/docs)
 - [pgx Documentation](https://pgx.github.io/pgx/)