While losing the cord would be a plus, we don't recommend using AirPlay or Chromecast to stream Zwift from your computer or mobile device to your TV. AirPlay and Chromecast are only intended for photos and video, and video games tend to lag badly when streamed.
We do recommend using either an appropriate video cable (HDMI, DVI, etc.) to connect your device to your external display. For iOS devices, there are a variety of Lightning to HDMI cables/adapters.
If you choose to use one of the streaming options instead, be aware that any issues with choppy video are unrelated to Zwift.