Perhaps the dreaded neutral safety switch on the tranny. It's almost always been my misfortune that 2 electrical faults are involved. My 4.5 used to do the no start till cool routine also.

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Present
1971 280SE W108

1967 250SE W111
Rhode Island
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Auf Wiedersehen
1972 280SE 4.5 (AKA Das Moneypitten)
1972 250C (Mit den zwei carburetors from hell)

"Time fly’s like an arrow, fruit flies like a banana" Groucho Marx