Plugin-registered web providers (registered via agent.web_search_registry)
were invisible to the tool-availability gate: _is_backend_available() was a
hardcoded env-var if-chain that returned False for any name outside the eight
built-in backends. Because check_web_api_key() is the check_fn for both
web_search and web_extract, a working custom provider with no built-in creds
left both tools filtered out of the toolset entirely.
Fix at the single chokepoint: _is_backend_available() now delegates non-legacy
backend names to the registered provider's is_available(), falling back to the
legacy built-in probes for known names and unregistered providers. Because
_get_backend(), _get_capability_backend(), and check_web_api_key() all resolve
availability through this one function, the fix cascades to every caller —
including the per-capability extract selection that produced a dead-end
'search-only' error (#32698). The two remaining hardcoded whitelist
early-returns (_get_backend, check_web_api_key) now also accept registered
names, and both walk registered providers as a final fallback so a custom
backend still resolves when no built-in has credentials.
Built-in backend priority is preserved unchanged: the registry is consulted
only for names outside _LEGACY_WEB_BACKENDS.
Fixes#28651Fixes#31873Fixes#32698
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