Hello! I found problems with the SystemChrome.setEnabledSystemUIOverlays ([]) method.
I will try to describe in as much detail as possible.

So, let's begin
I created an empty project and replaced it with this one.

Code

Code taken from here https://flutter.io/cookbook/navigation/navigation-basics/

import 'package:flutter/material.dart';

void main() {
  runApp(MaterialApp(
    title: 'Navigation Basics',
    home: FirstScreen(),
  ));
}

class FirstScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
   print(""" filterLog 
    <<<<<<  FirstScreen build >>>>>>>>
    """);
    return Scaffold(
      appBar: AppBar(
        title: Text('First Screen'),
      ),
      body: Center(
        child: RaisedButton(
          child: Text('Launch screen'),
          onPressed: () {
            Navigator.push(
              context,
              MaterialPageRoute(builder: (context) => SecondScreen()),
            );
          },
        ),
      ),
    );
  }
}

class SecondScreen extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    print(""" filterLog 
    <<<<<< SecondScreen build >>>>>>>>
    """);
    return Scaffold(
      appBar: AppBar(
        title: Text("Second Screen"),
      ),
      body: Center(
        child: RaisedButton(
          onPressed: () {
            Navigator.pop(context);
          },
          child: Text('Go back!'),
        ),
      ),
    );
  }
}

1. Normal State

Code unchanged.
Check the logs for comparing with the following items and note the movement of the contents inside the window when it is in the tray.

2. Full Screen on

Changes in the code.

1. Added method SystemChrome.setEnabledSystemUIOverlays ([]);
2. And Third Screen()

Please note that the widget that started the application does not start the assembly after minimizing the application. However, if you go to the second screen, and then to the third, it will call the build function on all the following screens, except the first, when we return to the application.
Note: In the following paragraph, we consider the option with Navigator.pushReplacement.

3. Full Screen on and change logic next Screen

Changes in the code. The Navigator.push method has been replaced by Navigator.pushReplacement, the second screen is now the main screen, but when we return from the tray, it still calls the build method.
Also note that the second screen is the main one, and when the “Back” button is pressed, the application is hidden and the widget calls the build method !!!!!! What for????? According to the logic of the android, by pressing back button on the root activity, the activity is killed

4. Hide bottom bar

Code changes instead of SystemChrome.setEnabledSystemUIOverlays ([]);
SystemChrome.setEnabledSystemUIOverlays ([SystemUiOverlay.bottom]);

When you open the application, everything is in order, the bottom panel is hidden, but if you tap the screen in any area, it will appear and disappear only after the application goes to the tray and back. The build method is still called after the tray.

5. Hide status bar

Code changes instead of SystemChrome.setEnabledSystemUIOverlays ([]);
SystemChrome.setEnabledSystemUIOverlays ([SystemUiOverlay.top]);

The top pane does not appear when you tap the screen (everything is fine), and not like in the previous example. The assembly method is still called when you go to the tray and back.

6. Full screen mode is enabled using native Android code.

Added the following code

package com.example.flutterbugsystemuioverlays

import android.os.Bundle
import android.view.View
import android.view.WindowManager
import io.flutter.app.FlutterActivity
import io.flutter.plugins.GeneratedPluginRegistrant

class MainActivity(): FlutterActivity() {
  override fun onCreate(savedInstanceState: Bundle?) {
    super.onCreate(savedInstanceState)
    window.setFlags(WindowManager.LayoutParams.FLAG_FULLSCREEN,
            WindowManager.LayoutParams.FLAG_FULLSCREEN);
    window.decorView.systemUiVisibility = (View.SYSTEM_UI_FLAG_IMMERSIVE
            or View.SYSTEM_UI_FLAG_LAYOUT_STABLE
            or View.SYSTEM_UI_FLAG_LAYOUT_HIDE_NAVIGATION
            or View.SYSTEM_UI_FLAG_LAYOUT_FULLSCREEN
            or View.SYSTEM_UI_FLAG_HIDE_NAVIGATION
            or View.SYSTEM_UI_FLAG_FULLSCREEN)
    GeneratedPluginRegistrant.registerWith(this)
  }
}

This is the most interesting case.

1. In all previous examples, the content inside has shifted when trying to hide the upper or lower bar
2. For some reason, the lower bar is not hidden, although this code works fine for native applications android
3. Also note that even though the top bar is hidden, the assembly function in the second screen is not called.

Flutter doctor -v

[✓] Flutter (Channel beta, v0.9.4, on Mac OS X 10.14 18A391, locale ru-RU)
    • Flutter version 0.9.4 at /Users/alexfix/Flutter/flutter
    • Framework revision f37c235c32 (6 weeks ago), 2018-09-25 17:45:40 -0400
    • Engine revision 74625aed32
    • Dart version 2.1.0-dev.5.0.flutter-a2eb050044

[✓] Android toolchain - develop for Android devices (Android SDK 28.0.3)
    • Android SDK at /Users/alexfix/Library/Android/sdk
    • Android NDK at /Users/alexfix/Library/Android/sdk/ndk-bundle
    • Platform android-28, build-tools 28.0.3
    • Java binary at: /Applications/Android Studio.app/Contents/jre/jdk/Contents/Home/bin/java
    • Java version OpenJDK Runtime Environment (build 1.8.0_152-release-1136-b06)
    • All Android licenses accepted.

[✓] iOS toolchain - develop for iOS devices (Xcode 10.1)
    • Xcode at /Applications/Xcode.app/Contents/Developer
    • Xcode 10.1, Build version 10B61
    • ios-deploy 2.0.0
    • CocoaPods version 1.5.3

[✓] Android Studio (version 3.2)
    • Android Studio at /Applications/Android Studio.app/Contents
    • Flutter plugin version 30.0.1
    • Dart plugin version 181.5656
    • Java version OpenJDK Runtime Environment (build 1.8.0_152-release-1136-b06)

[!] Connected devices
    ! No devices available

Conclusion

It turns out that at the moment it is impossible to use the usual full-screen mode for Android.
The native flutter method calls assembly methods for all screens, except for the first after returning to the applications from the tray.
In addition, the flutter somehow does not allow the native Android code to hide the lower bar, which would be a quick decision to remove the call for the assembly function on all screens, except for the first after the tray in full screen mode

I hope I somehow helped, because I really liked the flutter.