I'd suspect a poor connection somewhere right off the top. You *could* measure the voltage as you suggest and it may show okay but with the motor running with full load amps, a poor connection may have enough resistance to prevent the motor from running properly, it at all.
Improperly done factory crimp connections are not uncommon.
Testing with 12 volts directly at the motor or at the switch is a good diagnostic check and is probably what I'd be doing. Pulling the switch out is easy so probably a good place to start.
