Tavily web search always ran at Tavily's default depth (basic), because the search request never sent `search_depth`. The only Tavily depth control in Admin > Settings > Web Search, "Tavily Extract Depth", applies to the Extract API used by the web loader, never to search.
This adds `TAVILY_SEARCH_DEPTH` (env var and persisted setting, default `basic`) and a "Tavily Search Depth" select (ultra-fast, fast, basic, advanced) under the Tavily search engine settings. The value is sent as `search_depth` on every Tavily search request, so admins can set search and extract depth independently, for example fast search with advanced extraction.
The default matches Tavily's own default, so existing setups keep the same behaviour until the setting is changed.
Fixes#29891