Welcome! An interesting thought experiment. Just off the top of my head, I can think of a few reasons why this might be impractical. You’d have to power the PC, which would mean running a cable from the rig to the wall somewhere. I suppose this could be achieved with a large battery pack, but PCs with graphics cards consume quite a bit of power.
The computer would also have to be mounted somewhere that would be able to slide between games that are closely placed together. It could be mounted higher, but that would make the rig top-heavy or be distracting to the player. I thought for a moment about a mini-form PC, but it would have to be powerful enough to decode all the incoming video feeds.
I suppose it could theoretically be done with a gaming laptop like I use. The battery would last longer than trying to power a full PC, and it technically can function without having to use a full PCIe slot by use of HDMI capture cards. You could then control the whole thing from a remote desktop as you suggested.