What on earth are you talking about? A Bayesian estimator for peak likelihood, given a uniform prior on distributions, is what you described as "Dirichlet". It's one of the first examples you hit in many textbooks on Bayesian analysis.

Your comparison makes about as much sense as saying, "Nobody uses cars any more; we all use regenerative braking!"

You also mention that "Bayes" wholly lacks confidence intervals. That's ridiculous! You can do exactly what you stated with posterior distributions (it's usually called a credible interval, though). And as long as you've computed the whole posterior distribution, not taken a shortcut to find the mean or median or something, you can multiply it by a cost function too, if you want.

In fact, I'm not sure how one would even use the Dirichlet distribution for statistics outside of a Bayesian context. It makes a nice prior distribution.