test(api): harden provider policies and split search tests

[hanityx]

Summary

• Split provider policy tests and search-related tests into focused files.
• Harden provider constants, action safety, backups, and path handling.
• Colocate search tests around the modules they exercise.

Stack

Base: PR 1 provider extension groundwork.
Next: route and system boundary split.

Validation

• Full stack validation previously run: corepack pnpm qa:provider PASS, corepack pnpm qa:pr PASS.
• Final hygiene pass after docs/test fixture cleanup: api-ts tests 43 files / 273 tests PASS, TUI tests 69 PASS, public markdown hygiene PASS.