At least I think it's an air supply question... Our 2000 E320 has a warm starting issue. It starts fine when cold, and fine when hot. However, there is a range in the cooling down process that it just cranks.
I have run tests with two different gas caps, even without the cap, and find that after shut down at normal driving temp (80) it will restart fine until 1 1/2 hours have passed. Then it refuses to catch until another 15 minutes goes by. Then it starts fine.
The local mechanic tried the ring-gear lube trick to no avail. We are kind of surprised that the issue has not "thrown a code", but imagine there are odd intermittent malfunctions that the computer doesn't identify as serious.
I visualize a sticking butterfly valve (or weak spring) on top of the carburetor. Except there isn't one.
Any ideas?

scope the crank sensor when it doesn't start. my money's on that.

lubing the ring gear is a starter/flywheel issue and has only to do with cranking and nothing with the engine actually starting.

If it is on the original cps go ahead and replace it anyway.

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Jim