It doesn’t need storage permissions, if your phone isn’t allowing it to save, then it’s something that you have set up differently
I see, in that case I'm not sure what would be stopping you from accessing the save files, but nothing was changed regarding that in the last 10 versions, so it's 99% to do with your device.
There is a long way to back up your saves, though. You can transfer them to the web version via upload/download sync, and then export them there as a backup.
Importing them works the same way, just import to web -> upload sync -> download sync on the app