feat: add browser service shutdown to server lifecycle

apps/api/src/index.ts

@@ -385,6 +385,14 @@ async function shutdown(signal: string) {
     // AI package may not be available
   }
 
+  try {
+    const { shutdownBrowser } = await import("./lib/browser-service.js");
+    await shutdownBrowser();
+    console.log("Browser service shut down");
+  } catch {
+    // Browser service may not have been initialized
+  }
+
   try {
     await shutdownAnalytics();
     console.log("Analytics flushed");