Even with the sleeve screwed all the way into the neck of the case, the stem was too short to push the clutch into winding mode. You can make something shorter, but longer is a problem. It will have to be replaced.
The next image shows the tool used to adjust the depth of the sleeve spring in the neck of the case. It turns the sleeve in and out. It’s critical to have exactly the right size tool to avoid damaging the soft threads in the case. I have seen MANY cases pretty much ruined by stripping these threads, due to someone using a screwdriver or something. I have 9 or 10 of these tools with an assortment of sizes and styles of tips. And cases do vary that much.
By the way, most vintage American watches are “negative setting”. With this design, the “snap” in and out of the crown is a function of the case only, as it is here. This makes cases somewhat interchangeable. Modern watches are “positive setting”; the “snap” is a function of the watch mechanism, and the stem is part of the movement. That design is much more durable. These sleeve springs and wear out and break all the time.
This watch continued to fail to get through the whole testing cycle without a mysterious stopping. Simply touching it started it again so the problem was been hard to examine. I have narrowed it down to the escapement, which was perfect. I recleaned and readjusted it from top to bottom multiple times.
In (very rare) cases like this, the next thing I do is to simply replace perfectly good parts. I started with the escape wheel. Now I don’t have an electron microscope, but there was nothing I can detect wrong with this part. However Elgin used the same part for this for decades, so it’s not hard to just drop in another one in its place.
If it was a scarce part, I guess I would hand polish each face to a mirror finish first, and go from there.
