Open it up, take the PCB out, figure out how the dial sensor works (the dial is probably just repeatedly nudging a contact out of the way), and thennn... program an Arduino to be a giant input buffer (that slowly replays), and wire the Arduino between the dial and the rest of the machine. :D

Caveat: said Arduino may need to sit between the rest of the buttons as well, in case inputs must be given in sequence.

Possible benefit: switch the Arduino for something from the ESP family, and you could feed instructions to the machine over Wi-Fi or Bluetooth. (Which could take the form of a dedicated wireless button panel above the machine with macros for frequently-used options.)

(Hmm. I can tell there's some over-optimization in here, but I'm not sure _where_.)