The culprit was my pushbutton starter wire.

It ran from the battery to the purple ignition wire near the fuse box, and was quite hot when I touched it.

The previous owner had cut the ignition wire as part of an elaborate engine-kill alarm system. (Apparently, he was good friends with McGuyver and the A-Team.) Before I bought the car, he removed the alarm but left all his wires, relays and other crap behind, which I cleaned out a couple of years ago, during which time I had a "happy accident" that necessitated the pushbutton starter in the first place.

As I prepared to replace the existing pushbutton wire with a thicker, insulated wire, I discovered, concealed in a black sleeve of sorts, the other half of the above-mentioned purple ignition wire! I joined them together, turned the key, and the car is back to its normal self; no more pushbutton starts. (But I am already missing it. It was very cool.)

As a test, I let the engine run until it got auxiliary fan hot, then shut it down for a few moments let it get a bit hotter. Held my breath, crossed my fingers, turned the key, and it cranked and started like the happy car I remember.

On to the next problem...

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2008 E350 4matic / Black/Anthracite

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Gone but not Forgotten:
2001 E430 4matic, 206,xxx miles, Black/Charcoal
1995 E320, 252,xxx miles, Black/Grey
1989 260E, 223,00 miles, Black/Black