feat(chat): align model picker controls across frontends

[su-fen]

Summary

• remove the execution-mode timing hint from the GUI and WebUI model pickers
• add model and provider search to the WebUI model picker with automatic focus and empty results feedback
• add mirrored regression coverage for model picker search behavior

Validation

• node --test test/chat/execution-mode-model-picker.test.mjs
• pnpm exec tsc --noEmit in crates/agent-gateway/web
• pnpm exec biome check src/pages/chat/ChatHeader.tsx
• pnpm build in crates/agent-gateway/web
• git diff --check