It isn't all that clear what other users with similar symptoms means, nor have I seen any users suggest that making some sort of change to the HDMI cable changes the status of the AirPlay icon as discussed. I understand why you think what you are thinking but you keep using words like must without providing any evidence of any kind.
I don't think it's significant at all that the problem is seen on all devices on the network, this would be exactly the scenario if the problem/solution was in the Apple TV, the network, the network environment or the network router. I do believe the solution is most likely to be found in one of those areas and whilst I'm far from convinced it's anything to do with the HDMI cable, I do acknowledge its a possibility that it is, I do however feel it's rather a stretch to conclude it must be related to the HDMI cable.