Of course, this does in my view have the implication that it is appropriate for people who understand Bayesian language to use it when discussing their beliefs, especially in the context of a disagreement or other situation where one person's doesn't understand the other's thought process. I suspect this is the real point of controversy here (cf. our previous arguments about using numerical probabilities).

Yes, the reason why I often bring up this point is the danger of spurious exactitude in situations like these. Clearly, if you are able to discuss the situation in Bayesian language while being well aware of the non-Bayesian loose ends involved, that's great. The problem is that I often observe the tendency to pretend that these loose ends don't exist. Moreover, the parts of reasoning that are opaque to introspection are typically the most problematic ones, and in most cases, their problems can't be ameliorated by any formalism, but only on a messy case-by-case heuristic basis. The emphasis on Bayesian formalism detracts from these crucial problems.