If the machine was exposed to water, snow, rain, or a combination of all, there is a possibility of some damage to the ignition system, or the fuel supply, or anything that could have been affected by wind blown water, especially salt water!
Better to have another person look at it. Usually, one look-see by somebody will result in higher fix costs.
As for one cylinder running and one not--does he think the non-running piston just sets there, not moving? HMMM?? He's sounding expensive!
If the machine was under water for a time, there might be some water in the oil, which can be ascertained easily.
I';d say: "Pull the flywheel cover and check the ignition stuff around it. Clean the rust off the legs of the ignition coil and flywheel magnets. Blow out any dirt that has gotten in around the ignition areas.Blow out the mouse "hotel rooms". Check all safety switches for water penetration.
Clean the battery cable connections. Check cables for rust inside their covers. If you do get it to run a little, then change the oil! If you haven't done that already.
HTH: Rusty Jones, the Mower Man

Sound advice. Time to pull it apart. Its actually very well protected where its located and not subject to water which is good. But I wonder whether mice might be present and done some damage.
Hmm.

Can add nothing to Ruty's comments other than agree.

I've always found it very interesting that automatic standby generators don't charge the battery while running. The battery charging is done off AC when present. I recall one of the power outages we had here, my brother has a Generac "whole house" unit that he would shut down once a day to check the oil, etc. About the 5th time he did that it wouldn't start? Checking, he discovered the battery was low, "jumped" it off a garden tractor, and it started right up. He then ran an extension cord and plugged in a battery charger to charge the battery. We both thought it amusing that the generator was running fine, but he had to "back feed" to keep the battery up.

Guess the 120V feed coming in to charge the battery is also used to "sense" loss of power and begin the starting sequence? I know the manufacturers are forced to "cheapen" things to get to particular price points, but does a charging coil on a Kohler engine cost that much more than one without? Rhetorical, seems like all or most do it that way. Perhaps it's also the additonal circuitry and wiring to provide a sensing input and a charging input.

Just an observation,

Good luck,

Ev

Thanks Guys. Its coming apart in a slowly but surely. .