Illegal? No. Another rebroadcast system? Yes. Therefore subject to rebroadcast contracts and the Transmit Clause? Yes.
If you want to run a business that repeats broadcast TV content, no matter how you grab it, you need to pay the original broadcaster. You are narrowcasting a broadcast, and that is an add-on service.
Aereo was (is?) a sexy service. However that service still boiled down to getting subscribers to pay for something without paying the people that gave it to them. It's not private, personal use -- it's commercial, subscription use.
If you charge people to watch the game at your house, then you are violating most professional sporting organizations' rules about telecast reuse. Calling it a special, precious, individual use of the same sporting event only makes you a tongue-twister.
I know we're supposed to fetishize any new Internet service that sticks it to analog or clumsier versions of the same. Nevertheless, the contract rules are pretty clear. This doesn't throw out the 2007 Cablevision decision mentioned in the article -- it merely clarifies the requirements for the company providing its subscribers their personal show sets.
Think about how this differs from other Cloud-esque content. If I have a copywritten file that I put on Dropbox, it's available only to me wherever I am. If I try to share it, Dropbox will restrict that change in sharing automatically if they know about it (they have some neat checksum software).
No matter what, I found the content and put it in the Cloud myself in the above example. Aereo was putting the stream into your Cloud -- you weren't sending up your own recordings from the air. It all centered on rebroadcast: you get an antenna from Aereo, they select when to transmit someone else's content to it. They aren't sending you your holiday films -- they're sending you local TV channel 7.
This doesn't really set a precedent for all the other Internet startups to go under. You can do new things with existing stuff, but old contracts will still apply. The Supreme Court said the same thing in the 1840s, when they found that contracts signed before the USA broke off from the UK are still legally binding. You may give people what they want more effectively, but you'll still have to pay the owners for what you resell if they have contracts and lawyers to enforce that.
Note: I keep editing this as I sort out the exceptions.