It has 16 steps, 0-15k, and the output works just fine with a 4-bit binary TTL level input
I see no problem in the software side - you have a few input pulses, you steer a few output solenoids. Timings are not crucial, and the Arduino should be fast enough to poll/scan the inputs (you can extend the number of pins with shiftregisters or use the Mega, input or output)
A single Arduino is enough for en entire pinball machine, unless you want to drive a screen pixel-by-pixel (there is not enough RAM - but if it can be fed serially and "pictures" are stored in an SD card or such like ...). From your description it seems there is some logic in the driving board.
BTW: The point of the Scandivaian forum is either that we write in DK/SE/NO/SF or some aspect that is highly localized (where to buy, events in a city). But that is just a guideline, not a rule.