Just because voltage is good does not mean enough amps are getting through to turn over starter. I had this same issue several years back, good voltage, cleaned all connections but all I got was the clicking sound. I finally traced the cable from the genset to the house batteries and found the cable inside the terminal was slightly green even though the outside was clean and tight. When I wiggled the cable it was also slightly loose. After cleaning the cable end and terminal I soldered and crimped the cable end and 10 years later all is still well.