Yeah, exactly. It's so apparent to me that I figure something like this has to exist, if I dug into it.
I don't know if you'd even 10 relays. Most appliances have been cost-reduced into like two motors, a heater, and one or two actuators. Perhaps newer models with steam cleaning etc have reversed that trend.
Control side, I'm thinking of an AVR + ESP32. The AVR for reliably controlling the motors with the sensors, eg if the water is full, turn the water off. True safety stuff like overheat switches obviously stays inline though.
One of the more complex details is newer appliances have moved to brushless motors for efficiency. A three phase motor driver is straightforward, but figuring out what design parameters will be reasonably generic to serve the most appliances is some research.
The least generic parts seem to be fitting in with the existing user-input buttons, and managing the complexity of programming many different models for non-engineer users. The kind of people that are comfortable taking apart appliances generally have less patience for programming computers.