For what it's worth, Dropbox does have an alternative permissions model where the app is sandboxed into its own folder. I used it recently for a project of mine, and it was really easy to set up, so I highly recommend checking it out.
Yes, Uberzet has access to your entire Dropbox (as is made clear in http://uberzet.com/sharing-more it doesn't access anything other than your Public folder.
https://www.dropbox.com/developers/start/core