The first quoted analogy is close - you're leaving out that aereo only sells its service to people in that particular market. So, if you were in San Francisco, you wouldn't be able to listen to a NYC station online due to their service. This is one of your bigger misunderstandings.
This was never a misunderstanding. It has zero to do with the case at hand. The fact they are keeping it localized has little to do with them still doing more than simply amplifying a signal. They are taking in a signal from an antenna, recording it, and broadcasting that recording to users.
And your second analogy gets closer to your other misunderstanding. It's more like you downloading free software and burning it to a CD, and selling that CD to another user - and charging only for the convenience of having that put on a CD for a customer who may not be able to download it themselves due to a poor internet connection.
Selling the CD would be redistribution of the software. My second analogy might not be the best, as I was trying to avoid already established copyright laws such as redistribution, as it isn't exactly relevant to OTA broadcasts.
Why would they need more than one DVR per customer? Further, why would they need to be separate DVRs? It could be one giant, super-fast DVR; with them recording many copies of many shows simultaneously; provided there was a 1 to 1 customer to recording ratio.
A separate system, no. A separate process to record and redistribute? Yes. They are claiming 1 to 1 to the customer. Why are they even recording it? Do they offer "on demand" type services for this content or the ability to pause, rewind, fast forward? Why aren't they simply processing each individual signal into the format require for them to stream? That is one of the reason they were found guilty IMO.
The ruling was based on "similar to a cable company" basically saying they think their business felt like a cable company to them so they should doing what a cable company does which is pay for OTA.
Cable companies always capture OTA programming and always stream it constantly. A subscriber just changes the channel to the proper channel and starts watching the "live" stream of the OTA broadcast. Aereo technically collected no OTA transmissions nor recorded anything until a subscriber actually pushed "watch" on their particular antenna. It is a substantial difference in tech and business model that was completely disregarded by the majority but pointed out by the minority dissent.
Which is why the dissent, written by Scalia, said based on the ruling at SCOTUS that if Aereo has instead just recorded all OTA broadcasts but only allowed users of their service to watch the programming out of "sync" from real time of the broadcasts they would be "legal" according to the current ruling today with conjunctions to rulings made for cloud storage/streaming.
It wouldn't be legal. It would be separate "performance", and thus still copyright infringement, if I understand correctly. The dissent was that the direct infringement by Aereo isn't correctly expressed and simply using "cable company-like" is too confusing. It is even stated in the dissent he believed Aereo was breaking / circumventing the law, just that the Copyright Act should not be redefined in this case in the way it was.