I would go with 32 circuits, not counting servers or other house loads, I would put them on two separate panels, and I would use separate neutrals (no MWBCs). I kid you not; that is what I would recommend to the owner.
This is not about the amount of load, nor about minimizing the cost of installation. It is about the reliability of the power supply, and the continued availability of each and every item of gaming equipment. Under no circumstances would the owner of such an establishment want to have to shut down two systems, in order to work on the power source to one system. Every item is a revenue generator, and every item that is out of service does not generate revenue.