I'd highly recommend reading the ruling. It's a good read and the points are made more eloquently than I can put them. I'm not speaking as to whether or not the law is a good law, but this interpretation of the law seems pretty solid. Your hypothetical Dropbox scenario is actually addressed at the bottom of page 16, but you need context from earlier parts of the ruling to understand the "public" argument.
It's not really worth me repeating the argument here, as it would take 16 pages to do, and the Supreme Court has already done the work. After reading it I'd love to see your arguments with specific parts of the text.
My main beef is I feel they're stretching pretty far to claim the transmission requested by an individual from a private antenna to that individual is actually to the public. The argument I'm objecting to is around Section 3, page 12-14 or so. The transmissions were each individually requested by each user, and chosen to watch at unique times and durations by each user. That, to me, is sufficient to mark each transmission as private. None of the transmissions are identical or sent to anyone who did not explicitly request them, unlike the CATV operators they continually draw comparisons to. There is no part of that process that I think can reasonably be called a public transmission.
Page 16 does address the Dropbox scenario, but I don't think it provides a clear line, which makes this a pretty crappy opinion, in my opinion. As they say on page 16, the cloud operators could try to slip through the holes in their definitions of "perform" and "the public", but I don't really see the distinction. I, the user, used my legal right to place copyrighted work on the remote service, and I chose to play it back some time later. That sentence applies both to Aereo and to Dropbox, and I think this decision would make the Dropbox scenario illegal.
It's not really worth me repeating the argument here, as it would take 16 pages to do, and the Supreme Court has already done the work. After reading it I'd love to see your arguments with specific parts of the text.