iOS only supports AirPlay which is a proprietary Apple technology. Miracast is a standard for doing something similar, and is supported by Windows and Android, but it requires a direct wifi connection between the two devices, so the hardware in the phone/tablet and the TiVo would need to support direct wifi connections. (not sure the TiVo does) The Chrome mirroring uses a realtime video encoding, which requires a pretty beefy PC to accomplish. I know they added this to Android but I believe it uses the hardware encoder of the device, used for video recording, to function.
The functionality that allows you to cast a YouTube video or Netflix stream is very simple. It basically just passes a URL to the app on the TiVo and the app plays the linked stream. Other then that it's just a basic set of control commands that allow pause/play and seeking.
Developer for VideoReDo