Using MANAGE_EXTERNAL_STORAGE is not a viable solution, especially if you’re planning to publish your app on the Play Store, as it will likely get rejected. The proper way to save files on Android 34 is by using the MediaStore API. It doesn’t require any special permissions to write to public folders.
I’ve developed a Flutter plugin that solves this issue for you. Feel free to give it a try, and if you encounter any problems, you can open an issue on the repository. I’d be happy to help!