RE: PoE via GigE, according to Wikipedia it provides maximum of 25.5 Watts with up to 57 volts at 600 mAmp. Not sure how that corresponds to the power draw of a 5d3/1DX type camera. Other thing to think about though, is that they then need to add the charging circuit stuff into the camera, which adds a bit of complexity and cost to the camera. Not saying it's a lot, but it's something more to add. Maybe use PoE as an alternate power source alongside the battery.
PoE for tethered shooting is super interesting. The LP-E6 battery of the 5D3 is 7.2V and 1800 mAh. This means it's approx. 13 watt hours. From what I can see, since I don't really shoot video myself, battery life is about 90 minutes of video. Which means the camera draws somewhere around 8.7 watts for video.
So, the 5D3 for video would be no problem for PoE. Shotoing stills with AF and IS shouldn't be a problem either as there's quite a bit of room there as well.
Now, the 1DX, I don't know. It uses a 11.1 V battery @ 2200 mAh so it is 24.42 watt hours. I can't find any stats for the 1DX and video but it would probably be OK as well.
Now, charging and using the camera at the same time may be too much for PoE but just for use for external power and tethered shooting makes a lot of sense.