Common: Fix logging shutdown race condition.

Got a crash during logging on another thread during shutdown.

UI/NativeApp.cpp

@@ -1402,14 +1402,6 @@ void NativeShutdown() {
 #endif
 	g_Config.Save("NativeShutdown");
 
-	// Avoid shutting this down when restarting core.
-	if (!restarting)
-		LogManager::Shutdown();
-
-#ifdef ANDROID_NDK_PROFILER
-	moncleanup();
-#endif
-
 	INFO_LOG(SYSTEM, "NativeShutdown called");
 
 	ShutdownWebServer();
@@ -1420,6 +1412,14 @@ void NativeShutdown() {
 
 	g_Discord.Shutdown();
 
+	// Avoid shutting this down when restarting core.
+	if (!restarting)
+		LogManager::Shutdown();
+
+#ifdef ANDROID_NDK_PROFILER
+	moncleanup();
+#endif
+
 	if (logger) {
 		delete logger;
 		logger = nullptr;