Riverpod's provider doesn't store the state

riverpod_provider_doesnt_store_state.dart

import 'package:flutter/material.dart';
import 'package:flutter_riverpod/flutter_riverpod.dart';

void main() => runApp(ProviderScope(child: const MyApp()));

final counterProvider = StateProvider<int>((ref) {
  //final unsub = someStream.subscribe((val) => ref.state = va;);
  //ref.onDispose(() => unsub());
  return 0;
});

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      debugShowCheckedModeBanner: false,
      theme: ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: const MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  final String title;

  const MyHomePage({
    super.key,
    required this.title,
  });

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text(widget.title),
      ),
      body: Center(
        child: Row(mainAxisAlignment: MainAxisAlignment.spaceEvenly, children: [
          const CounterAndButton(),
          ProviderScope(
            overrides: [
              counterProvider.overrideWith((ref) => 4),
            ],
            child: const CounterAndButton()
          ),
          ProviderScope(child: const CounterAndButton()),
        ]),
      ),
    );
  }
}

class CounterAndButton extends ConsumerWidget {
  const CounterAndButton();

  Widget build(BuildContext context, WidgetRef ref) {
    final counter = ref.watch(counterProvider);
    return Column(
      mainAxisAlignment: MainAxisAlignment.center,
      children: [
        const Text(
          'Counter:',
        ),
        Text(
          '$counter',
          style: Theme.of(context).textTheme.headlineMedium,
        ),
        FloatingActionButton(
          onPressed: () {
            ref.read(counterProvider.notifier).state += 1;
          },
          tooltip: 'Increment',
          child: const Icon(Icons.add),
        )
      ],
    );
  }
}