Are multiple Arduino boards needed

Agreed. Without seeing the whole plan, it's hard to tell what's best for something of this size and potential complexity.
For long lengths between buttons (my laser maze wire length using 18AWG speaker wire goes about 40 feet wire length between three industrial deadman buttons and a central Arduino UNO) I think I would daisy chain Nanos if I had a way to power them semi permanently. I've had good success offloading tasks from a central Arduino to accessory Nano and just poll a digital pin for a HIGH/LOW signal, then do the thing at that junction.