How ψ-ontic are ψ-ontic models?

Ψ-ontology theorems show that in any ontic model that is able to reproduce the predictions of quantum mechanics, the quantum state must be encoded by the ontic state. Since the ontic state determines what is real, and it determines the quantum state, the quantum state must be real. But how does this precisely work in detail, and what does the result imply for the status of the quantum state in ψ-ontic models? As a test case scenario I will look at the ontic models of Meyer, Kent and Clifton. Since these models are able to reproduce the predictions of quantum mechanics, they must be ψ-ontic. On the other hand, quantum states play no role whatsoever in the construction of these models. Thus finding out which ontic state belongs to which quantum state is a non-trivial task. But once that is done, we can ask: does the quantum state play any explanatory role in these models, or is the fact that they are ψ-ontic a mere mathematical nicety?