P(sane things plus crazy things | speaker is saner) * P(speaker is saner) = P(speaker is saner | sane things plus crazy things) * P(sane things plus crazy things)

The fact that P(sane things plus crazy things | speaker is saner) <> P(speaker is saner | sane things plus crazy things) isn't a problem, if you deal with your priors correctly.

I think I misinterpreted your original question as meaning "Why is this problem fundamentally difficult even for Bayesians?", when it was actually, "What's wrong with the reasoning used by the speaker in addressing this problem?"