Just because you agree to do N doesn't mean you agree to do N+1 or N+2, etc. The networks do broadcast content, that's very expensive to produce, for free, in return for use of the public spectrum. That's the bargain. But the bargain doesn't mean that they agree to let other people do whatever "they see fit" with the broadcasted content. One part of that bargain, embodied in the copyright rules, is that you can't take the broadcast video and perform it publicly. I.e. I can't tape NBC shows and then open up a movie theater playing them. That's not part of the bundle of rights NBC agreed to give up in the process of broadcasting the show over the air.

All Aereo does is take advantage of a loophole. They couldn't record the shows and stream them, because that would violate the networks' copyright. So they try to shoehorn effectively the same service into the exception for time-shifting.