Why not? If you've got a hare to butcher, you'd likely want to split it. You could leave the carcass whole and roast it, but that's only one of the many ways to use the animal.
If you want to argue over details that make little difference though, I'd recommend using the term "splitting hairs" instead.
I was jesting but a serious reason why 'quote unquote' may be confusing is the fact that this kind of immediate negation is not considered correct grammar in any other context I know of.
I feel like writing quote-on-quote takes away the ability of non-native speakers to search for and learn the basis of the idiom. I really don't see how it would be more understandable than just typing "quote".
