Those are the devices I was thinking of. If my understanding of these devices is correct, they require a direct device to device connection and would not work with packet switching devices like a switch or router. That means you would need two Ethernet lines running from your office to your living room in order to get both usb and HDMI. In addition, you run into the bandwidth issue I was describing earlier where the HDMI signal caps out at a maximum of 1080p60 and it could be lower if the signal degrades enough over the long distance.

In my opinion your either better off going with a single fiber optic cable that can handle both usb and display (like thunderbolt or one of the newer usb standards) or using a cheap device that supports game streaming. Using a single cable is fairly expensive as the cable itself isn’t cheap and could require ancillary hardware. It could also necessitate snaking cables in your walls. As for game streaming the cost of entry can be pretty low as you primarily just need something that can decode the stream for the tv. An old laptop (I’ve even had some success with an old acer chromebook) or even a phone might be sufficient for you to give game streaming from your pc a try to see if you like it and if the latency is sufficient. My vote would be to cobble something together for free to try out game streaming to see if you like it and then go from there. RPi3’s are capable of running game streaming and don’t break the bank.