"WRITE_EXTERNAL_STORAGE is deprecated (and is not granted) when targeting Android 13+. If you need to write to shared storage, use the MediaStore"

As of Android 13, if you need to query or interact with MediaStore or media files on the shared storage, you should be using instead one or more new storage permissions:

• android.permission.READ_MEDIA_IMAGES
• android.permission.READ_MEDIA_VIDEO
• android.permission.READ_MEDIA_AUDIO

Looks like your device is running Android 13+ and that's why you cant request "WRITE_EXTERNAL_STORAGE"

MANAGE_EXTERNAL_STORAGE supersedes WRITE_EXTERNAL_STORAGE & allows broad access to external storage. You don't need WRITE_EXTERNAL_STORAGE if you already have MANAGE_EXTERNAL_STORAGE permission granted.

To publish apps with MANAGE_EXTERNAL_STORAGE permission, you'll need clear justification for requesting it, otherwise the app will be rejected. Since you're not publishing the app on Google Play, it's fine. But those who think MANAGE_EXTERNAL_STORAGE is the alternative to WRITE_EXTERNAL_STORAGE, they're wrong. If they can't clearly justify the need to request MANAGE_EXTERNAL_STORAGE (which most apps can't), their apps will be rejected. Their choice should be either app specific external storage (Android/data/package-name) access or the ones described in documentation above