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".
'kind unkind'
'known unknown' (unless your Donald Rumsfeld)
'seen unseen'
etc.