Hi, well the shore power transfer switch is under the operational control of the generator AC output. When the 120VAC from the generator is on the timer in the transfer switch starts its cycle (typically 30 seconds delay after start of generator) and then the timer board will connect power to the transfer relay coil to operate the switchover from shore contacts to generator output contacts. The transfer relay removes the shore power connections (it does not matter if shore power is connected or not) and connects the generator output connections to the coach AC panel and its loads. When you manually operate the transfer relay you are proving that the generator output can be connected to the coach AC panel.
The manual link I previously enclosed says (I think) that when the generator starts and sends 120VAC to the timer board, after the 30 seconds delay the LED light on the timer board should be lit indicating that it is sending 120VAC to the transfer relay coil. If the LED does not light after the generator starts plus about 30 seconds delay, the control board is not working correctly. Note that the transfer relay coil could also be bad.....but start by replacing the timer board. You may find that it is cheaper (under $100) and easier to just replace the entire transfer switch (age of coach?) and then keep the working parts of the old one for future emergency spares. If you are not familiar with AC circuits be careful to work on the transfer switch with gen and shore power off or get a licensed electrician to help you.