That is actually not true. If your AppleTV has tvOS 9 or newer, you can activate P2P AirPlay. It’ll create an Ad-Hoc network only between your TV and Mac/Phone and use that for AirPlay, no further network or internet connection necessary. You just have to activate it in your AppleTV settings.

Edit: Alternatively, the Hollyland Mars 4K is a pretty decent wireless video transmitter that supports HDMI, that I’ve used in a professional setting sometimes. Not sure it is what you’re looking for (or sensible, price wise) but you probably won’t get a more stable connection with anything cheaper.