The engine started from cold but was hesitant: it started to try to fire but required continued starter engagement to get going. It idled very slowly for 5-10 seconds then accelerated to its fast idle (I have ordered an idle control box). It idled very smoothly until
it warmed up and I turned the key off. After a couple hours I trued it again and the performance was the same: hesitant start then up to fast idle circa 1000 rpm.

Cold start function and repeated dying and dtarting, but it runs great until it starts dying suddenly

The common element in difficult cold starting and sudden shut-off is the fuel pump relay.

Did you ever run the simple on-the-car fuel accumulator test I referred you to in post #11? It's in a thread started by me...easy search.

Duke

I have a new fuel accumulator to install when I have time to do it.
One interesting thing about the run-die episodes: it involves temperature. Engine runs perfectly until it suddemly dies. Restarts get more difficult as it continues to heat up. 5 minutes engine off with the hood up restores engines ability to run for a few miles, and the more it cools down, the longer it will run before dying. Seems a typical electronics failure. My ‘86 420 SEL did this when the crank position sensor on the flywheel failed; this engine seems to have only a front sensor for tdc.

I recallthat, on the ‘86 420, the ignition module was mounted on the steel fender liner which served as a heat sink. There was heat transfer paste between them and it was said to dry out and lose effectiveness so I removed the module, cleaned out the dried-up paste, and replaced it.
Is this true for the module on the 560 SL?
After replacing the fuel pump relay, it still hard-starts, like it needs a choke, but runs perfectly for 2-3 miles then dies suddenly. It will restart immediately but die within 20 feet, but the longer I wait ( and let it cool), the longer it will run after restarting, so the fault is obviously temperate-related.
Hard starting cold, like it needs a choke. I replaced the idle control module and cleaned the idle control valve but no change. The struggle continues...
If it seems like I don’t know what I am doing, well...

Randy, you should apply heat transfer paste to the back of the ignition control module. It's important that it be there. An electronics store should sell the paste. Those ignition control modules cost thousands of dollars. Don't want to fry one.

The 1986-1991 W126 and 1986-1989 R107 share a common engine management system. The EZL is more or less the same design, and yes heatsink goop is required. Lack of it will not cause the EZL to overheat and die as quickly as you're experiencing though.

When you have a stall condition, you need to determine if you have spark. If you fail to do so, you're farting into the wind for any other troubleshooting.

If you do have spark, you have a fuel problem. Pumps, fuel pump relay, ignition switch, fuse box, wiring, etc.

If you do not have spark, you have a problem with the OVP relay, EZL, crank sensor, ignition coil, or distributor. For what it's worth, a failed crank sensor will also cause the fuel pumps not to run after the initial prime.

One step at a time. Inspect items and deem them good or bad then move on. Firing the parts cannon at the car isn't going to fix the problem. If it does, it's sheer luck and will cost a LOT more than following good troubleshooting procedure.

1988 560 SL UPDATE
 

Car has been in the hands of a professional for almost one year. He has replaced the fuel distributor, several other parts, cleaned out the fuel tank more thoroughly than I was able to do. Car is running beautifully, except it dies repeatedly after warm. He is looking for a crank position sensor and there seems to be none anywhere to be had.
After my experience trying to find parts for this car, I got discouraged and sold the 1986 420 SEL I have owned since late 2003. I have a feeling this car is next; it's a beauty but, with no parts, how can I live with it?

Get a can of cold spray. Elecronic components store a good source. Just before it quits timewise. Spray the suspect component. Or just give it random shots, To keep the temperature low. Keeps running you have it. You can usually figure out a way to test any component. Car quits at ldle after a period could help a lot. In spraying components.

Early heavy v8 Mercedes had a strong igniton system I thought. I remember some issues with their coils. That one may be newer that the ones I am thinking about. If the car will quit at idle. Keep cooling the coli.

By now someone has had to establish it is an ignition or fuel issue, Sounds like it has been decided it is ignition. A year by a guy that knows what he is doing? Have to wonder a little.