Post edit.
The 1971 cite was not meant to speak to origin, just to date. It was the earliest cite I’ve seen.
2½" was hung like a shrew, that’s why they should have hanged him 
I’m not hot under the collar, just annoyed that you’d nitpick something that’s not all that important to the OP, who wanted to know about the morphological-semantic nature of the verb. The expression hang someone out to dry is not slang. It’s an idiom. You’re looking in the wrong dictionary. When a literal expression becomes an idiom, OED rarely explains when and how it happened. OED is long enough as it is.
Laundry is neither dead nor alive (OP’s concern). When you hang the laundry out to dry, you don’t abandon it; you take it back in eventually.
An animal, however, can be either dead or alive, and one that has been hung up to dry is dead, hence the analogy to someone who is no longer considered important.
That’s probably why the Cambridge Dictionary of American Idioms went with that etymology. You don’t have to believe Cambridge Press if you don’t want to.
But if it will make you happy, I’ll recant my statement. Instead,
“I THINK the origin of the expression comes from hanging a dead animal out to dry after it’s been killed.” Okay?
In any case, there is a literal use of the phrase in OED, though it’s intransitive: “hang (II.8.c) Of flesh for food: to be suspended up in the air to dry, mature, or become ‘high’.”