I could care less means that the speaker/writer could conceivably care less than they do at this moment.
So . . . they DO care. At least a little. Because they COULD care less.
So it's possible for them to care less than they do right now.
Yeah, not even ALMOST what they mean.
What they mean is that they care so little that there is no way for them to care any less than they already do. There is no caring here. At all. None. Zip.
Ergo, they could NOT or couldn't care less.
John Cleese says it better. He's also fun to watch while ranting about it: