https://en.m.wikipedia.org/wiki/Aereo
The problem is that this is the business model of broadcast networks in a world of cable TV. The statutory retransmission fees funnel some cable TV revenue back to the broadcasters. It doesn't really make sense from first principles, but it has been a pragmatic compromise.
The messed up thing about Aereo is that after being shot down by the Supreme Court, they tried to get a license to operate as a cable company but were shot down on that front, too.
My guess is the RF tech is a lot easier to deploy when you can plan for some percentile of peak utilization, rather than 1:1 with the user base. Even then, they'd undoubtedly face legal challenges, so they must have made the judgment call that the odds were on their side. Which may have been a reasonable conclusion. A lot of the judgments were actually on their side until the circuit split and the final Supreme Court decision.
After all, one wouldn't imaging that leasing an old-school aerial antenna would be a legal issue.
It was a hot topic on HN several years back. I may be biased, as my wife did PR for Aereo through the firm she works for. But I definitely buy the logic that what they were doing was technical a private performance.
This approach would make sense if the games are on paid channels and the nominal fee for those only covers viewing for one person (in which case it makes sense for establishments to pay more, given they're showing it to more people).
But if the games are broadcast on free channels, then there shouldn't be any difference whether or not a person is watching it at home or a bar is showing the game - in either case, the TV advertisers are the ones paying for the game, and this revenue depends on how many people watch the channel (so showing off the game in a bar actually benefits everyone).
I feel like in this situation it's the latter, and the people from the football league are just greedy (what a surprise) and any excuse to try and extort money is a good one in their book.