Of course Bayesians can look at the residuals! And of course there are bad models in Bayesian analysis. Maybe a few Bayesians in the 70's supported views like that (and I doubt that), but you will hardly find any Bayesian supporting this view these days.
I didn't read the text, but Bayesians use things like Bayes factors to compare models. Actually, a Bayesian can even compute the probability of a model being true and pick the model which is more likely to be true. Or a Bayesian can average across models, to achieve a better model. Or can use posterior predictive checks. There are a lot of options to check a model and each one may favor one approach or another, but to say that there are no bad models in Bayesian analysis is non-sense.
So, at most, it would be more appropriate to say that in some extreme versions of Bayesianism (extreme versions that almost no one uses in applied settings, by the way) you're not allowed to check your model. But than you could say that in some extreme versions of frequentism you're not allowed to use observational data as well. But why waste time discussing these silly things, when we can discuss if and when, in an applied setting, we should use Bayesian or frequentist methods or whatever? That's what's important, in my humble opinion.
Update: The OP asked for a reference of someone advocating the extreme version of Bayes. Since I never read any extreme version of Bayes, I can't provide this reference. But I'd guess that Savage may be such a reference. I never read anything written by him, so I may be wrong.
ps.: Think about the problem of the "well-calibrated Bayesian" (Dawid (1982), JASA, 77, 379). A coherent subjectivist Bayesian forecaster can't be uncalibrated, and so wouldn't review his model/forecasts despite any overwhelming evidence that he's uncalibrated. But I don't think anyone in practice can claim to be that coherent. Thus, model review is important.
ps2.: I like this paper by Efron as well. The full reference is: Efron, Bradley (2005). "Bayesians, frequentists, and scientists." Journal of the American Statistical Association 100(469).