Their argument isn't that they should be able to use the data freely because it's broadcast over the public airwaves. The relevance of the public airwaves is that there is an existing compromise between broadcasters and the public such that the public is entitled to watch certain content for free over the air, in return for broadcasters being allowed to use the public airwaves. What Aereo is arguing is that their service falls within this existing compromise, because it's no different than attaching a rabbit-ear to your TV with a really long cable.

In other words, it's okay not just because the data is on the public airwaves, but because the data is on the public airwaves AND broadcasters have agreed to let the public use that data under certain conditions. This isn't quite the legal posture of the dispute, but it's the public policy essence of it.