"It goes from a high idle to a real low idle and may die. Even if it doesn't die before landing, when the throttle is opened it coughs and dies then. It is not rich, it just dies."

This is lean a lean engine being described, not a rich one. It's quiting because it's running out of gas. Fatten the low speed up (1/8-1/4 turn?), adjust the idle speed so it idles at a decent rpm and try that?

The high idle/low idle is almost always being caused by a hot combustion chamber (unless you're talking about a DLE). That temp itself is not the problem. When you shut the throttle down there's not enough fuel coming in to cool the combustion chamber down, and it's not coming down until it cools? That's the problem... Yes, it comes down eventually, but you can skip this waiting period by running a richer low speed mixture that will throw more fuel in there to cool it down - right now.

When it comes to cooling, the fuel mixture is as critical a factor as any other!