feat: wire chat UI to Responses API with streaming

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>

src/ChatAgent.Api/Program.cs

@@ -16,6 +16,15 @@ var builder = WebApplication.CreateBuilder(args);
 // for explicit structure -- each controller is a separate file with clear routing (D-05).
 builder.Services.AddControllers();
 
+// Register a named HttpClient for proxying requests to the Responses API.
+// The base URL comes from appsettings.json (server-side config, not exposed to the browser).
+// IHttpClientFactory manages the underlying HttpMessageHandler lifetime.
+builder.Services.AddHttpClient("ResponsesApi", client =>
+{
+    var baseUrl = builder.Configuration["ResponsesApi:BaseUrl"] ?? "http://localhost:8317";
+    client.BaseAddress = new Uri(baseUrl);
+});
+
 // AddCors() registers Cross-Origin Resource Sharing services.
 // CORS is REQUIRED because the Blazor WASM client runs on a different origin
 // (https://localhost:5200) than this API (https://localhost:7100).