Some of those Kohlers were very sensitive to the gas filter. Did you replace it with a factory Kohler gas filter? How about the direction orientation? Either of those can cause the problems you discussed.

Make sure the vent hole in the gas cap is not obstructed. Little bugs love little holes.

I second the fuel filter idea.

Did it happen to die when the engine was being worked hard (mowing tall grass, going uphill, etc.)?

If you never put a big load on the engine to draw the highest amount of gas through the filter, you may never have a problem with those filters.

It does tend to have the problem a little more when going up hills, but not always.

Had this issue:
1:Changed the filter . did not work properly

2: removed the hose and blew into the gas tank , same issues, run and die.

3: rmoved the hose from carb and blew with hose this helped ,till changes gas.. it ran for like 10 minutes.

4: my gas cap did not have any air holes, so i just did not tightened al the way down...ran till i finished it takes a good hour to cut the grass...

Thank you Herb