Fix app-server TUI MCP startup warnings regression (#16041)

This addresses #16038

The default `tui_app_server` path stopped surfacing MCP startup failures
during cold start, even though the legacy TUI still showed warnings like
`MCP startup incomplete (...)`. The app-server bridge emitted per-server
startup status notifications, but `tui_app_server` ignored them, so
failed MCP handshakes could look like a clean startup.

This change teaches `tui_app_server` to consume MCP startup status
notifications, preserve the immediate per-server failure warning, and
synthesize the same aggregate startup warning the legacy TUI shows once
startup settles.

codex-rs/tui/src/app/app_server_adapter.rs

@@ -108,6 +108,19 @@ use codex_protocol::protocol::TurnStartedEvent;
 use std::time::Duration;
 
 impl App {
+    fn refresh_mcp_startup_expected_servers_from_config(&mut self) {
+        let enabled_config_mcp_servers: Vec<String> = self
+            .chat_widget
+            .config_ref()
+            .mcp_servers
+            .get()
+            .iter()
+            .filter_map(|(name, server)| server.enabled.then_some(name.clone()))
+            .collect();
+        self.chat_widget
+            .set_mcp_startup_expected_servers(enabled_config_mcp_servers);
+    }
+
     pub(super) async fn handle_app_server_event(
         &mut self,
         app_server_client: &AppServerSession,
@@ -119,6 +132,8 @@ impl App {
                     skipped,
                     "app-server event consumer lagged; dropping ignored events"
                 );
+                self.refresh_mcp_startup_expected_servers_from_config();
+                self.chat_widget.finish_mcp_startup_after_lag();
             }
             AppServerEvent::ServerNotification(notification) => {
                 self.handle_server_notification_event(app_server_client, notification)
@@ -146,6 +161,9 @@ impl App {
                 self.pending_app_server_requests
                     .resolve_notification(&notification.request_id);
             }
+            ServerNotification::McpServerStatusUpdated(_) => {
+                self.refresh_mcp_startup_expected_servers_from_config();
+            }
             ServerNotification::AccountRateLimitsUpdated(notification) => {
                 self.chat_widget.on_rate_limit_snapshot(Some(
                     app_server_rate_limit_snapshot_to_core(notification.rate_limits.clone()),