What I love about this BBS is that if somebody asks a technical question, you're pretty much guaranteed that it's a non-trivial problem to fix.

So it seems to me that the hypothetical product that you want is a USB-C KVM switch. Of course, such a thing doesn't exist yet. What does seem to exist are DisplayPort and HDMI KVM switches. (Here's a are several multi-port multi-monitor HDMI models, of course, not exactly cheap...)

As an alternative, you could probably play games with screen-scraping / remote-desktop software (e.g., VNC). Not really useful for gaming, but perfectly useful for managing a fleet of test gear or whatnot.