This sounds like a confusion between a theoretical perfect Bayesian and practical approximations. The perfect Bayesian wouldn't have any use for model checking because from the start it always considers every hypothesis it is capable of formulating, whereas the prior used by a human scientist won't ever even come close to encoding all of their knowledge.
(A more "Bayesian" alternative to model checking is to have an explicit "none of the above" hypothesis as part of your prior.)
NOTA is addressed in the paper as inadequate. What does it predict?
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