Change the custom App Icon implementation

### Description

At the moment, our custom icon implementation results in 2 app icons being shown. This has resulted in confusion and users uninstalling the app by mistake.

It would be great if we could investigate how other apps which support custom app icons like DuckDuckGo and ToDoList have implemented this.

Note: The JP Mobile team looked into this too; you can view their comments here: https://github.com/wordpress-mobile/WordPress-Android/issues/14476#issuecomment-1451884262

### Step-by-step reproduction instructions

1. Go to Appearance and set a custom App Icon
2. Notice you will now see two app icons for Pocket Casts

### Screenshots or screen recording

### Did you search for existing bug reports?

- [X] I have searched for existing bug reports.

### Device, Operating system, and Pocket Casts app version

Samsung Z Fold 4, Android 13, Pocket Casts v7.43

Start by reproducing the duplicate-icon behavior from Appearance on Android 13 using the custom App Icon setting. Review the existing custom app icon implementation and the linked WordPress Mobile discussion, comparing the approaches used by DuckDuckGo and ToDoList. Done means selecting a custom icon leaves only one Pocket Casts app icon visible.

Written by the indexing model from the issue text.