What about those of us who use multiple devices, or multiple browsers? I've been using local storage for years and it's definitely hampering adoption, especially for multiplayer.
I never tried it, but from the descriptions I have read, Dropbox detects conflicting file saves (if you save on two devices while they are offline) and stores them as "conflicting copies". So the user can handle the conflict.
As a developer, you would do this in the application. "Hey, you are trying to save your data but the data on disk is newer than when you loaded it ... Here are the differences and your options how to merge.".
You're suggesting an actual API-facilitated data sync via Dropbox? Sure, but at that point why? Unless the data also needs to be read by 3rd party applications, might as well host it myself.