Add ChatController that proxies POST /api/chat to the local Responses API
(localhost:8317/v1/responses) with SSE streaming. Client reads tokens via
SetBrowserResponseStreamingEnabled and renders them incrementally. Includes
thinking indicator, input disabled during streaming, and error handling.
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
