To be fair, Aereo did try to frame the argument that way. I wish I had a good citation, but when the case arrived at the court, the justices chose to hear the question the broadcasters posed over Aereo's. The former is very much framed as a copyright issue, whereas Aereo's preferred framing was to set themselves up as an equipment rental service. So, unfortunately (from my perspective), the arguments were couched in the broadcasters' semantics from the start.

What's so weird about the case is we're talking about Aereo providing, but not paying for, what's already freely available. That would hardly be controversial if not for the existing compromise with the cable companies. It would seem that they're following the letter of the law, but I didn't get the impression that the justices care about that distinction much.