Manually start the generator and check for 240VAC between terminals E1 and E2 in the transfer switch with the generator running. (Transfer switch can be in either position, voltage should still be there.)
Trouble can be in the transfer control circuit, which it looks like you have a good handle on in terms of troubleshooting, or in the generator not delivering power. The control logic from the relay uses the power from the source to which the transfer will occur to energize the actual transfer solenoid. In other words, the transfer-to-generator coil gets its power from the generator and the transfer-to-utility coil gets its power from the utility. Each coil is in series with a microswitch that cuts power to the coil immediately when the transfer is complete. Those coils pack a lot of punch to make the transfer extremely fast but they are designed for VERY intermittent duty.