feat(sidecar): investigate search-API-capable providers (e.g. Gemini) as web-search sidecar backends

[lidge-jun]

Follow-up split from #398 (sibling of #414).

Context

Beyond adding dedicated search vendors like Exa (#414), several LLM providers expose their OWN grounded-search / web-search capability that could back the sidecar directly using the account the user already has — most notably Google Gemini (Grounding with Google Search / google_search tool), and potentially others (Perplexity, xAI Grok live search, etc.).

Ask

Investigate which currently-supported providers can serve as a web-search sidecar backend via their native search API, and enable the viable ones:

• Gemini: does the Grounding-with-Google-Search tool return citations/sources we can map into SidecarOutcome (text + sources)? What auth (API key vs OAuth) and quota semantics apply?
• Enumerate other providers with a usable server-side search tool and note feasibility.

Deliverable

A short capability matrix (provider -> has native search API -> citation shape -> auth) plus enabling the clearly-viable backends behind webSearchSidecar.backend.

Relationship

• feat(sidecar): add Exa and other search providers as web-search sidecar backends #414 adds dedicated search vendors (Exa) as backends.
• This issue evaluates/enables LLM-provider-native search APIs (Gemini first) as backends.
• Both preserve the [Bug] Web Search/Vision Sidecar Backend Fixed to OpenAI·Anthropic - Routing Turns Fail with 499/502 When Limits are Exhausted #398 graceful-degradation contract.