From the way you described this occurred, I would tend to suspect contacts (circuit breaker switches and relays) over connection problems. Fortunately, this problem is limited to the generator power and power switching system, and therefore there aren't too many places to check. Once you get to the transfer switch the current routing is common with the shorepower, that works. You are probably going to have to take readings with an digital voltmeter, or maybe you could just use a non-contact voltage sniffer on the 2 phase hot wires. Fuses need to be double checked with an ohmmeter, not just a visual check. Circuit breaker contacts can go bad. The biggest thing that I would suspect would be the transfer switch relay contacts, due to possible arc pitting and corrosion. You MAY be able to get this to work with DeOxIt electrical contact cleaner/restorer spray on the relay contacts, or file the contacts with a burnisher or flat swiss file. Best to replace the relay if it is found to be the problem. Be sure to remove ALL voltage sources when working on the transfer switch relay.
A possible technique to PREVENT transfer switch contact damage is to work it so that when the generator comes on there is no electrical load present. You can do this by turning OFF the inside mains circuit breaker before starting the generator. Once the generator is up and providing voltage through the transfer switch, then turn ON the inside mains breakers. This puts all of the load connection electrical contact arcing on the inside mains breakers that are cheaper than the transfer switch, more accessible and easier to install than the transfer switch, and are available at most any hardware store. I keep an extra inside dual mains breaker on hand. When turning OFF the generator, similarly turning OFF the inside mains breaker while the transfer switch relay switches under NO ELECTRICAL CURRENT LOAD will prevent a similar electrical arc on the transfer switch relay contacts. Then turn the inside mains breakers back ON for shorepower.