In the old days the wash controller was a glorified electric clock or multifunction electromechanical timeswitch. These wore over time and caused trouble and one particular trouble was the pump running too long without water so overheating and eventual fire was the result.

Like most things, the manufacturers decided it would be better to use a cheap computer instead of making the controller better and so now the timing and switching functions are carried out by a small programmed computer circuit board making decisions about what action to take depending, like the computer in your car, on information from various switches and sensors.

While in theory any failed electronic components can be replaced, the means of discovering what has failed is beyond all except the most skilled engineers and so in practice the circuit board is considered to be a single throw-away component which is cheaper and more satisfactory in the long run than attempting any but the most basic of repairs - who ever heard of a computer motherboard getting repaired?

Some repair guys will have a stock of usable salvaged spare parts to help keep older machines functional for longer, that is the helpful sort of repair guy. Failing that you could check the cost of a new part on the internet and perhaps try ebay or Amazon...

Good luck!