/////It is only true that D flat is enharmonic to C sharp if you are playing stuff like Schoenberg's First Chamber Symphony (which does actually require that). The circular arrangement is making an assertion about tonal relationships that only applies to certain types of music./////

Urm...I would say that the vast majority of music requires that Db be enharmonic to C#. Anything you're going to use a piano or a guitar on is going to require that. I can't comment on Shetland music or anything Scottish--I have zero knowledge of non-American music outside of classical--but every music theory class and lesson I ever took assumes C# and Db are enharmonically equivalent. Anything that doesn't, well, sorry, but I did make clear in the opening post that we're not interested in that stuff here.