Does the pump run at all when using a timed irrigation program. I suspect it does not. Is it 120v or 240v motor? What model motor?
Turn off the breaker for the pump. Either a 120 or 240 volt breaker. Unscrew and remove the rear cover on the pump to inspect the wiring. Look for any burned or cut wires. A test light only proves there is power present. If it is 240 volt you have two hots and one leg could be dead (open). Burned contacts at the start relay could cause one leg to open.
Does the motor get hot at all when power is on? Seized bearings will lock the rotor and all motors have a lock rotor rating so as not to catch on fire. Usually you will not hear anything. I had a pool pump seized and had no idea until I noticed the water didn't look clear and when I checked the pump, the motor was very hot. It did not trip the breaker.
Most pump motors have start and or run capacitors and when they go bad the motor may not start if the run cap is bad. Some motors have a set of centrifugal contacts at the back of the motor. When the motor spins up it disconnects the start capacitor. It also can prevent the motor from running. Plenty of youtube videos on what a bad start cap looks like. You'll see oil or the cap is swollen, depending on the type.
Please be careful checking the voltage at the motor. Remember you're on damp ground and a short to a hot lead can be deadly.
