Thanks for the informative answer. It just occured to me that modern phones like iPhone or Motorola/Samsung/etc phones often have Qi charging capabilities for powering accessories, so powering a display wirelessly shouldn't be a problem.
The only issues I can think of there are I'm not sure if the Qi process can start up without power on the receiving end, the source device isn't always broadcasting power afaik so I don't know what the trigger is and if it requires some communication with the receiver device. The last is the NFC has a pitiful data rate of ~424 kbits/s but if we're powering it through Qi we can use more power hungry transfer methods to speed it up.