Yes.

OTOH, the expected value of the beta distribution with parameters a and b happens to equal the mode of the beta distribution with parameters a - 1 and b - 1, so maximum likelihood does give the right answer (i.e. the expected value of the posterior) if you start from the improper prior B(0, 0).

(IIRC, the same thing happens with other types of distributions, if you pick the ‘right’ improper prior (i.e. the one Jaynes argues for in conditions of total ignorance for totally unrelated reasons) for each. I wonder if this has some particular relevance.)