> it seems nobody have tried
Probably because it's too obvious of an idea? And you wouldn't just find people's repos on github with their ledger databases for another obvious reason.
It's like a ton of time to get into such framework, much easier to install some app on your phone. What's the point? (Sincere question)
I'm not sure it's worth a writeup as anything special, git is (for programmers especially) a conventional way of keeping things synchronized between devices. I also used Dropbox for a time, but I specifically wanted versioning to help keep track of edits and reorgs of my files so I switched to git.