Welcome to the club!
Kohlers are notorious for this problem. I have more or less the same generator
(a 230 volt 6.5EFOZ) and have had exactly the same problems.
Before you do anything, you should pull the heat exchanger and remove bits of impeller from it. You should also pull the exhaust elbow and make sure it's not blocked.
Then you need to deal with what is almost certainly an airlock. You have to burp and burp and burp the hoses, keep patiently refilling, until you get the air out. What I do after this is start it, and put an infrared thermometer on the flange above the fresh water pump. This is where heat will go up fast if coolant is not circulating. Feel the hoses to the heat exchanger. If those hoses get hot and the flange stays cool, you're in business. If not, however, then shut the gennie down before it shuts itself off, let it cool a bit, then start burping hoses again. It is tedious but it eventually works.
If you've got goo in the fresh water side (I had this too) then you've either got a bad heat exchanger letting salt water
into the fresh water side, or somewhere down the road someone put pink antifreeze
in after having had green antifreeze
. Are you losing coolant?
If you are not losing coolant, then I would flush about 20 times (and every time you do it, you have to repeat the burping/infrared thermometer ritual). Then put in new pink antifreeze and forget about it.
If you are losing coolant, then this adds up, I think, to a leaky heat exchanger, which can easily happen if you fail to replace the pencil anode in time. This is what happened to me, and I'm about to replace the heat exchanger, which is not cheap