However, the phrase makes no sense.

It's an idiom, I can give you a great many examples of idioms that don't make logical sense. I'm sure there are plenty of others that are similar to could care less - and you seem to understand it perfectly well, albeit begrudgingly. Why?

Because idioms don't have to make logical sense, a bare minimum I suppose is that it's syntactically correct and understandable by a large majority of any given language/region. Why is could care less an exception? There are plenty of idioms that supposedly don't make sense, yet this one is by far the most criticised. Is it because it's relatively recent?

Idioms, a lot of the time, are very contextual. If your argument is that "could care less" doesn't make sense in any instance, then you're quite clearly wrong. In instances where "couldn't care less" can be used instead? Also wrong.

Don't get me wrong. I understand perfectly well why people are so critical of other people's language - and perhaps sometimes people genuinely don't understand. Some people can harbour an awful superiority complex in regard to language usage, especially to nonstandard dialects and very informal language, they may very well have been taught grammar that is entirely superfluous - not splitting infinitives for example. But if you understand and can comprehend easily what another person is saying (regardless of the logical process involved in the literal gloss of an idiom) then what is the problem?

Yes, could care less is counterintuitive, nobody's disputing that - but if that's your only argument then by your standards the majority of the English language doesn't make sense. Languages as opposed to what you might think, do not evolve intuitively, giving rise to the most efficient possible discourse. Quite the opposite actually, there are so many factors that decide how any given language will change and evolve that it's nigh impossible to predict what the exact future of a language will be - so there is no point in viewing things like this as a logical, intuitive process - it isn't. Not could care less, not whom or who, not split infinitives, your/you're, they're/their/there or any of the other purportedly logical vices of English. I don't even want to get into orthography, that makes the least sense.