No, they can't; don't be silly. That's not what "promoting" means, and in any case such a use would be a clear copyright violation (note elsewhere in the license where they make it clear that you still own the copyright). It seems clear to me that the intent is things like the Youtube "recommended" list. If you upload a public video, they need permission to repackage it for display in other contexts than direct viewing.

But if you insist on reading the license that uncharitably, you have to accept that Dropbox too can find plenty of wiggle room to be evil. Seriously, you're granting "all needed permissions" without enumeration -- you don't find that scary? What if Dropbox decides that "providing the service" requires revenue gained from scraping credit card numbers from stored files? See? I can come up with equally silly scenarios.