Would be cool if you could still work offline and sync designs once you’re connected again.

Not just in this case, also for when traveling or working in places with no internet connection.

We are missing quite some files in our Design Organization. Specifically from after May 25th. It looks like they are restoring backups but haven't replaced it fully yet.

Edit: This was a major assumption. Not sure what happened as of yet, but some of our recent files had been moved to the drafts folder of the person who initially created the file.

Does their desktop app still work, or is it simply a wrapper for the website?

It's basically a wrapper.