Before I would take the cover off of the transfer switch, I would plug in to shore power again and see if it clicks. If it clicks, the transfer switch is OK.
There is no need to do that again with the generator running.
If the transfer switch does not click when you start the generator, check the circuit breaker on the generator itself. These may be difficult to find on some generators but it should be on the front side. Try resetting the circuit breaker and see if that fixes your problem. If not, it may be time to contact someone to service your generator.