Yeah, an 802.11 AC network is pretty much required 
The only scenario that 802.11n would work is if you were in the same room as the access point, as you described. Wifi generally slows down real fast the further you move away from it. And since 802.11n’s realistic throughput is low, by the time you’re too far away from it, you’re going to have speeds way below the 16-20mbit bitrate that live TV streams use.
Live TV streams are very high bitrate, so you need a network that can really push it.
Again though, there’s not much more variable than wifi speeds. They vary from house to house due to many many factors. So what might work ok for some people might not work for others.
My best suggestion would be to get the SpeedTest app and test what kind of speeds you are getting on your device in a specific place in your home. It’s important to remember though, that the SpeedTest app tests your internet speeds. So if your internet connection is slower than your network, it wouldn’t be a great way to test it.