After trying for many hours, I've come to the conclusion that there's no way to backup, export or move an existing Minecraft world (where one may have spent hours, days or weeks on) once it has been created in the "application" storage (default setting).

I find this quite a huge issue, considering that in Android 13+ there's no way to access the ~/Android/data/* folder unless you have access to the root user, and rooting usually involves wiping your phone data; thus I have decided to report it as a bug, even though it technically isn't.

One that may have spent a lot of time on the game may want to make sure that his Minecraft worlds are backed up elsewhere, for example in multiple hard drives or in cloud storage, he may decide to start running the world on PC 24/7 so that his friends can join while he is away, or simply, he may want to change his phone and port his Minecraft world to his new phone.

Regardless of the case, it's only reasonable to expect Minecraft to have a simple "Export to External Storage" option which copies a world from the Application storage to the External storage.