I think the court has found that the "distribution for pay", i.e. "Cable TV" aspects of aereo's service outweighs the "antenna rental" aspects, and that makes it charging for a public performance.

If you rent an antenna on top of the Empire State Building, digitize the signal, and set up your own means to stream it over the internet to your house in Peoria, that is probably a private performance. If you effectively sell someone access to the stream (like a Cable TV company), that is a public performance.

This is also what ESB would be doing by letting people rent antennas and streaming servers.

Aereo is selling streaming TV service that just happens to use antennas in the backend.

You might say, it's all the same in the end from a technical point of view. And you're right, but the law doesn't care about the technical point of view. From the legal point of view, Aereo is selling TV streaming, and must comply with laws for TV streaming.