Duet 2 is the right choice, it brings pins out for external drivers. Looks like there are 5 drivers, plus some other stuff for the heated bed and the endstops.
Assuming that diagram is reasonably accurate, you should be able to keep just about everything while replacing the controller board. The only question is the button panel. The ones for emergency, and lighting, that do not go through the controller, those should work. There are a couple that perform actions, and those will be easy to set up. I'd be very surprised if the touchscreen ever works with a duet. You might even want to remove it or tape over it.