The “click” is it’s solenoid closing okay, but apparently full amps aren’t flowing through due to some resistance. Something like this leads me first to check the heavy ground connection at the generator and battery post(s), then check that full 12+ voltage is on both sides of the generator’s starter solenoid’s large posts when one of the switches is pressed ON. If not, then the solenoid’s contacts may be corroded from non-use.
That said, I’ve not had issues of this sort with ours, so others here may correct my presumptions, as I’m not familiar with exactly how the Genny starter is configured.
Joel