Having non-global or circular preferences doesn't mean a utility function doesn't exist - it just means it's far more complex.

Can you expand on that? I can't find any description on the web of utility functions that aren't intimately bound to global preferences. Well-behaved global preferences give you utility functions by the Utility Theorem; utility functions directly give you global preferences.

Someone recently remarked (in a comment I haven't been able to find again) that circular preferences really mean a preference for running around in circles, but this is a redefinition of "preference". A preference is what you were observing when you presented someone with pairs of alternatives and asked them to choose one from each. If, on eliciting a cyclic set of preferences, you ask them whether they prefer running around in circles or not, and they say not, then there you are, they've told you another preference. Are you going to then say they have a preference for contradicting themselves?