Flutter App with Firebase Hosting including Authentication and Database - flutter

Question

I have created a small Flutter app, which includes firebase auth and database. I want to deploy it with firebase hosting and did all the necessary steps, but it is still not working. The webside is just empty or MAYBE waiting for something (not sure about that).
firebase login
firebase init
added "site": "xxxxxxxxxxx-d1d99" to my firebase.json file
added js sdks to index.html:
<script src="/__/firebase/8.3.2/firebase-auth.js"></script>
<script src="/__/firebase/8.3.2/firebase-database.js"></script>
firebase deploy
Also I have chosen the 'web' directory instead of 'build/web' as my public directory, because my flutter project creates the flutter create web there.
My main.dart looks like the following. Except from the MaterialApp(), it is completely the same as suggested by flutter/firebase on how to do it.
main.dart
import 'package:faschingsplaner/views/auth/authentication.dart';
import 'package:faschingsplaner/views/auth/root_page.dart';
import 'package:firebase_core/firebase_core.dart';
import 'package:flutter/material.dart';
import 'package:flutter_localizations/flutter_localizations.dart';
import 'views/add_carnival_view/add_carnival_view.dart';
import 'views/home_view/home_view.dart';
Future<void> main() async {
WidgetsFlutterBinding.ensureInitialized();
runApp(MyApp());
}
class MyApp extends StatefulWidget {
#override
_MyAppState createState() => _MyAppState();
}
class _MyAppState extends State<MyApp> {
// Set default `_initialized` and `_error` state to false
bool _initialized = false;
bool _error = false;
// final Future<FirebaseApp> _fbApp = Firebase.initializeApp();
// Define an async function to initialize FlutterFire
void initializeFlutterFire() async {
try {
await Firebase.initializeApp();
setState(() {
_initialized = true;
});
} catch(e) {
setState(() {
_error = true;
});
}
}
#override
void initState() {
initializeFlutterFire();
super.initState();
}
#override
Widget build(BuildContext context) {
if (_error) {
return Center(
child: Text('Error occured: $_error'),
);
}
if (!_initialized) {
return Center(
child: CircularProgressIndicator(),
);
}
return MaterialApp(
debugShowCheckedModeBanner: false,
localizationsDelegates: [GlobalMaterialLocalizations.delegate],
supportedLocales: [
const Locale('de'),
],
theme: ThemeData(
brightness: Brightness.dark,
primaryColor: Color.fromRGBO(58, 66, 86, 1.0),
accentColor: Colors.blue),
initialRoute: '/authentication',
routes: {
RootPage.routeName: (context) => RootPage(auth: new Auth()),
HomeView.routeName: (context) => HomeView(),
AddView.routeName: (context) => AddView(),
});
}
}
At this point, I have totally no clue what could be wrong. Please consider that I am pretty new to flutter.