Hi @gagarin and @dyneq I was able to finish up a very rough schematic for the control board and other inputs in fritzing. Please excuse the wiring, I'm not familiar with the part editing so had to use the layouts I could find. The big chip is indeed the MCP23S17 (not the MCP23017 as on the graphic labeled).
Anyway, the main trick consists in sharing the 3 serial lines (SCK, MOSI, and MISO) between all 3 chips, then using separate digital pins from axoloti to the NSS/CS pins of each chip to enable them. The rest is basic potentiometer wiring, some series resistors for the LEDs and seven-segment displays, and a couple switches and encoder quadrature. I may have flipped a few pins from the axoloti serial protocol, so make sure you try each piece on a breadboard before soldering anything up!
(Note: the outputs from the MCP23S17 are arranged for diagramming convenience - probably a good idea to wire them according to the standard ABCDEFG 7-segment schematic so you can use a standard truth table, although I ended up making my own by trial and error anyway)
(Edited to add some decoupling capacitors.)