improvements(example): fix router navigation issue and add improvements

flutter/example/lib/auto_close_screen.dart

@@ -0,0 +1,47 @@
+import 'package:flutter/material.dart';
+import 'package:sentry/sentry.dart';
+
+/// This screen is only used to demonstrate how route navigation works.
+/// Init will create a child span and pop the screen after 3 seconds.
+/// Afterwards the transaction should be seen on the performance page.
+class AutoCloseScreen extends StatefulWidget {
+  const AutoCloseScreen({super.key});
+
+  @override
+  AutoCloseScreenState createState() => AutoCloseScreenState();
+}
+
+class AutoCloseScreenState extends State<AutoCloseScreen> {
+  static const delayInSeconds = 3;
+
+  @override
+  void initState() {
+    super.initState();
+    _doComplexOperationThenClose();
+  }
+
+  Future<void> _doComplexOperationThenClose() async {
+    final activeSpan = Sentry.getSpan();
+    final childSpan = activeSpan?.startChild('complex operation',
+        description: 'running a $delayInSeconds seconds operation');
+    await Future.delayed(const Duration(seconds: delayInSeconds));
+    childSpan?.finish();
+    // ignore: use_build_context_synchronously
+    Navigator.of(context).pop();
+  }
+
+  @override
+  Widget build(BuildContext context) {
+    return Scaffold(
+      appBar: AppBar(
+        title: const Text('Delayed Screen'),
+      ),
+      body: const Center(
+        child: Text(
+          'This screen will automatically close in $delayInSeconds seconds...',
+          textAlign: TextAlign.center,
+        ),
+      ),
+    );
+  }
+}