Fix: CupertinoSheetTransition moves SystemUIOverlayStyle to outside of delegatedTransition and only changes top bar (#164680)

Fix: CupertinoSheetTransition should set the SystemUIOverlayStyle in
init state
fixes: #164633 
Fixes https://github.com/flutter/flutter/issues/164134

## Pre-launch Checklist

- [x] I read the [Contributor Guide] and followed the process outlined
there for submitting PRs.
- [x] I read the [Tree Hygiene] wiki page, which explains my
responsibilities.
- [x] I read and followed the [Flutter Style Guide], including [Features
we expect every widget to implement].
- [x] I signed the [CLA].
- [x] I listed at least one issue that this PR fixes in the description
above.
- [x] I updated/added relevant documentation (doc comments with `///`).
- [x] I added new tests to check the change I am making, or this PR is
[test-exempt].
- [x] I followed the [breaking change policy] and added [Data Driven
Fixes] where supported.
- [x] All existing and new tests are passing.

packages/flutter/lib/src/cupertino/sheet.dart

@@ -258,8 +258,6 @@ class CupertinoSheetTransition extends StatefulWidget {
     final Animation<double> scaleAnimation = curvedAnimation.drive(_kScaleTween);
     curvedAnimation.dispose();
 
-    SystemChrome.setSystemUIOverlayStyle(SystemUiOverlayStyle.light);
-
     final bool isDarkMode = CupertinoTheme.brightnessOf(context) == Brightness.dark;
     final Color overlayColor = isDarkMode ? const Color(0xFFc8c8c8) : const Color(0xFF000000);
 
@@ -350,6 +348,12 @@ class _CupertinoSheetTransitionState extends State<CupertinoSheetTransition> {
   @override
   void initState() {
     super.initState();
+    SystemChrome.setSystemUIOverlayStyle(
+      const SystemUiOverlayStyle(
+        statusBarBrightness: Brightness.dark,
+        statusBarIconBrightness: Brightness.light,
+      ),
+    );
     _setupAnimation();
   }
 

packages/flutter/test/cupertino/sheet_test.dart

@@ -4,6 +4,7 @@
 
 import 'package:flutter/cupertino.dart';
 import 'package:flutter/material.dart';
+import 'package:flutter/services.dart';
 import 'package:flutter_localizations/flutter_localizations.dart';
 import 'package:flutter_test/flutter_test.dart';
 
@@ -1212,4 +1213,38 @@ void main() {
       expect(finalPosition, equals(initialPosition));
     });
   });
+
+  testWidgets('CupertinoSheetTransition handles SystemUiOverlayStyle changes', (
+    WidgetTester tester,
+  ) async {
+    final AnimationController controller = AnimationController(
+      duration: const Duration(milliseconds: 100),
+      vsync: const TestVSync(),
+    );
+    addTearDown(controller.dispose);
+
+    final Animation<double> secondaryAnimation = Tween<double>(
+      begin: 1,
+      end: 0,
+    ).animate(controller);
+
+    await tester.pumpWidget(
+      CupertinoApp(
+        home: AnimatedBuilder(
+          animation: controller,
+          builder: (BuildContext context, Widget? child) {
+            return CupertinoSheetTransition(
+              primaryRouteAnimation: controller,
+              secondaryRouteAnimation: secondaryAnimation,
+              linearTransition: false,
+              child: const SizedBox(),
+            );
+          },
+        ),
+      ),
+    );
+
+    expect(SystemChrome.latestStyle!.statusBarBrightness, Brightness.dark);
+    expect(SystemChrome.latestStyle!.statusBarIconBrightness, Brightness.light);
+  });
 }