Writing your own dishwasher firmware is a little hairy, there is safety to think about with leaving a heater or fill solenoid on. You have to filter and debounce all inputs, and have timeouts if a switch or float gets stuck, pump doesn't work, that kind of thing. But nothing an Arduino couldn't pull off

The controller board looks straightforward, I would see if it's possible to read the dishwasher MCU's firmware and disassemble it.
Just need to post PC board pics and sketch a schematic to assign I/O, the MCU seems to have only 16 pins.