Flutter catching all unhandled exceptions

I'm trying to catch all unhandled exceptions in a Flutter app so I can sent it to a crash reporter. There are instructions on how to do this in the Flutter docs. I followed those, and added two bits of code to my app to catch exceptions:

Catch Dart errors by wrapping runApp in runZoned:

runZoned<Future<void>>(
  () async {
    runApp(MyApp());
  },
  onError: (dynamic error, StackTrace stackTrace) {
    print("=================== CAUGHT DART ERROR");
    // Send report
  },
);

Catch flutter errors by setting FlutterError.onError:

FlutterError.onError = (FlutterErrorDetails details) {
  print("=================== CAUGHT FLUTTER ERROR");
  // Send report
};

However, when I test this at runtime by throwing an exception from a button:

throw Exception("Just testing");

The exception appears in the console:

════════ Exception Caught By gesture
═══════════════════════════════════════════════════════════════
The following _Exception was thrown while handling a gesture: Exception:
Just testing When the exception was
thrown, this was the stack:

… etc

But I see no sign of my print statements (CAUGHT DART ERROR or CAUGHT FLUTTER ERROR), and setting breakpoints on those lines never seems to hit, so I think my exception handling code isn't catching it. Am I missing something?

Here's a minimal reproducible example (click the button, which throws an exception, but it's not caught as expected):

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

void main() =>
  runZoned<Future<void>>(
    () async {
      runApp(MyApp());
    },
    onError: (dynamic error, StackTrace stackTrace) {
      print("=================== CAUGHT DART ERROR");
      // Send report
      // NEVER REACHES HERE - WHY?
    },
  );

class MyApp extends StatefulWidget {
  // This widget is the root of your application.
  @override
  _MyAppState createState() => _MyAppState();
}

class _MyAppState extends State<MyApp> {

  @override
  void initState() {
    super.initState();

    // This captures errors reported by the FLUTTER framework.
    FlutterError.onError = (FlutterErrorDetails details) {
      print("=================== CAUGHT FLUTTER ERROR");
      // Send report
      // NEVER REACHES HERE - WHY?
    };
  }

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      home: Scaffold(
        body: SafeArea(
          child: RaisedButton(
            child: Text("Throw exception"),
            onPressed: () {
              throw Exception("This is a test exception");
            },
          ),
        ),
      ),
    );
  }
}