This is good advise.
The coils in the transfer switch that cause the contacts to switch are actually operated by 240VAC through rectifier bridges packaged in the coil assemblies. 240VAC utility power is needed in order to switch from the generator position to the utility service position and 240VAC generator power is needed in order to switch from the utility service position to the generator position.
Without the generator producing 240VAC power, there will be no connection made to the generator. The voltage on the conductors from the generator can be measured in the transfer switch to determine if the 240VAC is there.
If the generator has started automatically and 240VAC is present on the power conductors from the generator, the first thing to check would be the control relay in the transfer switch. It should be energized and the 240VAC from the generator should be routed to the transfer contactor coil. If there is 240VAC across the coil and the contactor has not moved, the coil or the rectifier in it could be bad.
Skip and Leroy, thank you for the advice. Our next planned trip to the cabin is January 14. I will check these things out when I do another test. Hopefully it will act up again.