I'm not sure about the overall goal. Do you want to have the ability to display the output of one gaming system to a central big screen? Is each game system stand alone with its own controller and only the sound and video needs to be switched to the central monitor?

If this is the case, each game system would need to have a two way split -- one output to it's dedicated monitor and the other goes to central.

A large matrix would be one approach with all of the game outputs run to a central point. From an administrative and future proof standpoint this would be the obvious approach, but at a cost.

A contrarian approach would be to use smaller, less expensive HDMI switches, local to a small cluster of games. Each cluster output would feed another switch, a cluster of clusters, if you like. Eventually, the final switch in the chain feeds the central monitor. In this scheme you would need fewer long HDMI cables. A control system can tie all of this together. The operator would simply indicate which game to display. You'll need to push some numbers around to figure out the most cost effective configuration. For example, 4-port switches are cheaper than 8-port switches, but you would need fewer 8-port switches and probably somewhat longer HDMI cables for 8-port. All of this must be balanced against the cost of one large, expensive central switch and HDMI wiring.

Note that switching from one game system to the next would not be instant with this multi-switch approach because there would need to be a chain of EDID negotiations. A central switch could offer instant switching (not all switches offer this feature) -- at a price.