So it is quite easy to check, and no need for complicated knowledge.
From my practical experience of debugging such systems, most likely faults were due to either current sensor, pressure sensor or relay failure.
First of all, how system knows that motor is OK? it either checks the current consumption of a motor, or there is a pressure sensor next to motor, so when motor is on, pressure change is detected. We need to rule out both situations.
We know the location of motor, right? So it should be possible to connect small incandescent light bulb to parallel to it. So after it is done, we should check the following, as power is applied to system and moment, where motor should start spinning.
Does lamp come on or not?
If it does, so should do the motor, let's check motor too. If motor is ok, proceed to the next next step.
If it does not, this means that there's a fault in a way of the power, we need to find it. To do this, disconnect the board from mains, disconnect the motor and apply small DC voltage, say 3 volts, from battery or whatever, to motor terminals. Using the multimeter, trace the voltage backwards, to see from where it comes. At certain point, you should "meet" the relay or other kind of switching device, past which your battery current won't go, since device is in offline stage. After you have identified that device, you need to check the voltage on it's contacts while system is operating (do this carefully!) Motor should be connected as usually it is. So when system is started and motor should be on, if there's a voltage (usually, full or almost full mains) on contacts of that switching device, it is not operating properly, since it should be in engaged state and show zero voltage drop on it. So the switching device needs to be replaced, but still, there are some chances that it simply does not receive the statement to execute. Similarly, if there's a half mains voltage or whatever on open contacts of a switching device, it still might be faulty and needs replacement.
But what if there's no voltage at switching device at all? Most likely, there's a blown fuse somewhere on the way of the power line, so you should again disconnect from the mains, and continue to trace the power delivery path, but this time, bypassing the switching device. Once you see the interruption on the power delivery, here you are, here's the faulty component which needs to be replaced.
Another issue is, when motor spins, but system believes it does not. This usually happens either to faulty pressure switch or current sensor. The pressure switch can be easily checked - usually it is next to the motor, in line with piping and has two contacts, which usually make contact under the required pressure only. So just connect multimeter to these contacts (disconnecting the wires going to it),and check for continuity - it should change when motor starts. If not, pressure sensor is faulty.
There are other, obvious possible causes of the problem, like clogged piping, but I guess, these were already ruled out?