An unfinished upload or sync stored locally instead of on the remote can absolutely be an issue. You can look at all the posts about OneDrive and GDrive not actually syncing before confirming to users who then delete their files since they "have been uploaded". Or the user may never open that specific page again or the session will not exist anymore when he comes back.

Browser storage is cheap, but it is not guaranteed to be durable.

That's a good point. You'd want some (overridable) way to block navigating away until the sync completed, or do it in the background. Local storage is pretty easy, remote can get tricky.