Some of those similar threads and posts might have suggested you have a clogged fuel tank cap vent hole. I suggest you try probing the cap vent holes (inside & outside) of cap to eliminate that as the cause.
Since you say this is quite predictable, you may want to try this test before cleaning the cap.
Run the tractor until it quits. Immediately remove the fuel tank cap and put it back on. Try to start the engine.
If the engine restarts after loosening and tightening the cap......clean out the vent holes in the cap.
If the engine does not restart after loosening and tightening the cap you will need to check for spark to the plugs when the engine is in this non-running condition (cranking but no spark).

Yip, I find the Kohler Command fails the coils in the 4-7th year. Almost on que. The common ways they fail are:

1. They run fine, turn the engine off, no cranky uppy until cools off.

2. Runs then starts backfiring and dies, no cranky uppy till cools off.

3. Just fails, nothing at all when cold.

I see these 3 quite regularly.

Like mownie said, start with the easy, then progress to the harder stuff if that won't solve the problem.

Thanks, Waiting for a service manual to arrive to order & replace the ignition coil. Got the point of no spark after running for 20 or 30 minutes.

Tired of waiting for the postman to bring your snail mail documents?
Then try the latest innovation in document distribution!
THE Internet (compliments of Al Gore).

Click the link below. Plug in the data for YOUR engine and voila............you get a free download of the service manual.
And you can give your postman the razzberry!

Here is a link that might be useful: Kohler manuals webpage