Cupertino gallery app (#2715)

Resolves #2519 

Part of the larger effort to bring the samples/repo into a stable and
maintainable condition that _also_ remains useful. See: #2409

## Pre-launch Checklist

- [x] I read the [Flutter Style Guide] _recently_, and have followed its
advice.
- [x] I signed the [CLA].
- [x] I read the [Contributors Guide].
- [x] I have added sample code updates to the [changelog].
- [x] I updated/added relevant documentation (doc comments with `///`).

cupertino_gallery/lib/widgets/picker_page.dart

@@ -0,0 +1,28 @@
+import 'package:flutter/cupertino.dart';
+
+class PickerPage extends StatelessWidget {
+  const PickerPage({super.key});
+
+  @override
+  Widget build(BuildContext context) {
+    return CupertinoPageScaffold(
+      navigationBar: const CupertinoNavigationBar(
+        middle: Text('Picker'),
+      ),
+      child: Center(
+        child: SizedBox(
+          height: 200,
+          child: CupertinoPicker(
+            itemExtent: 32,
+            onSelectedItemChanged: (int index) {},
+            children: const <Widget>[
+              Text('One'),
+              Text('Two'),
+              Text('Three'),
+            ],
+          ),
+        ),
+      ),
+    );
+  }
+}