I have owned an appliance repair business for a while now. There are a number of common cases.
Some controllers have a plug to dip switches that set the model that the board is installed in. In this case the board can be used across a range of products. All of the products will be of the same type, all dish washers, cloths washers, dryers, etc.. GE fridge & cloths washer controllers are a good example of this.
Then there are controllers that share the same PCB but only have the parts populated for the model that it is intended for. You see this a lot in wall mounted ovens. The controller in a single oven unit has only half the parts compared to one for a double oven.
Of course there is case where the controller is built with a single purpose in mind.
I worked in IT for many years and retired at an early age. I would see no reason why an Arduino or PI could be easily used in place of. A modular unit could be built to plug in the proper relays and input devises. Most of the manufactures controllers are large so it wold be possible to substitute in the current space.
If I were going to start an appliance manufacturing business, I would definitely look at doing something more uniform across models and products.