The Apple standard for its built-in hardware-accelerated displays, makes them suitable for full-motion video for production/display of cinema-quality video with NO dropped frames, and NO dropouts or partial-blank scan lines due to memory under-runs or other issues. This requires a hardware rasterizer/display-generator for each fully-accelerated display, supported by Huge memory bandwidth to refresh each display 60 or more times a second.
An Airplay display is NOT a hardware accelerated display, and is NOT a good fit for full motion video and similar activities. you are connecting across links with FAR too little bandwidth for full motion video.
If you are doing ONLY program listings, spreadsheets, stock quotes and other slow to change data, there are some other solutions, including AirPlay, but they require you to make some strong compromises. In most cases you really DO need a regular display on a high speed cable under a meter long.