The 365 was trouble from the get-go. Cadillac dealers replaced many of these motors under warranty for their customers, along with cylinder heads on motors that could still be salvaged. An older friend had a Cadillac dealership back in the day, and the mere mention of a 365, or the 1980's HT4100, will make him spit on the ground, followed by some choice words.
The same fellow also stated that as the 365 cars went out of warranty, many would be traded back to him with dead engines, for which of course he gave near nothing in trade in value. These cars were then repaired with 390 engines, which were somewhat plentiful in salvage yards and re-sold to others looking to buy a good used Cadillac. The 390 upgrade was posted on the windshield of these cars as a selling point, and it was!
Cadillac was not alone in having a dud engine.. the first Packard V8 which came out in 1955 , was notorious for losing oil pressure due to cavitation of the oil pump. The problem was evidently corrected for 1956, but it was too late for Packard.
Having attended numerous car shows, the one thing I have seen is that very few cars with the 365 retain their original engines.. many have been upgraded to the 390, and oftentimes, their owners are oblivious to that respect.