[codex] fix(web): honor registered provider availability

[stardust-mem]

Upstream value: Makes web-backend detection registry-aware so plugin-registered providers are no longer falsely reported as unavailable — a fix for the provider-plugin architecture that affects anyone using custom or third-party search backends.

Summary

Fixes #31873 by letting the web tool availability gate recognize provider names registered through agent.web_search_registry instead of only accepting the built-in backend list.

Changes

• Accept configured third-party web providers in _get_backend() when they are present in the registry.
• Delegate unknown backend availability checks to the registered provider's is_available() method.
• Make check_web_api_key() consider configured web.backend, web.search_backend, and web.extract_backend provider names before falling back to legacy built-in auto-detection.
• Add regression coverage for a fake kagi provider proving custom providers can expose web_search / web_extract, and that unavailable explicit custom providers do not silently fall back to another backend.

Validation

• pytest tests/tools/test_web_tools_config.py::TestBackendSelection tests/tools/test_web_tools_config.py::TestCheckWebApiKey -q
• pytest tests/tools/test_web_providers.py tests/tools/test_web_providers_brave_free.py tests/tools/test_web_providers_ddgs.py tests/tools/test_web_providers_searxng.py tests/tools/test_web_providers_xai.py tests/plugins/web/test_web_search_provider_plugins.py -q
• pytest tests/tools/test_web_tools_config.py -q
• python -m py_compile tools/web_tools.py
• git diff --check

[kshitijk4poor]

Superseded by #57779, which fixes this same bug class (plugin-registered web providers invisible to the tool-availability gate) as a single chokepoint in _is_backend_available() on current main — cascading to _get_backend(), _get_capability_backend(), and check_web_api_key().

Your PR targeted the same gate and informed the approach — thanks for the contribution, credited in the merged PR body. Closing as superseded. #57779