Bayesian methods help you to manipulate the data with priors, higher level models with a lot of overfitting, and poor identification of parameters, and yet it is not too noticeable. But if you do not engage in these practices, then it is essentially the same as maximum likelihood.
How useful is Bayesian econometrics?

When theorists build models of rational agents, they always assume agents are fully Bayesian. Because Bayesianism is the only rational course of action.
When applied people do applied work, they are almost never Bayesian. I wonder why? Maybe they are not rational?

No, you are talking about using conditional probabilities, instead of marginal probabilities, this is something all statisticians do, among other things a regression is relevant because it is a centrality measure of a conditional probability. Updating probabilities with new info is rational.
However, Bayesians do something different. They estimate parameters using a probability distribution (the posteriori) and any draw of this distribution is an estimator (in practice, they often take the mode of the posteriori) which is basically the ML estimator with a penalty determined by the prior. That is an inference procedure, and has nothing to do with rationality.

Bayesian methods help you to manipulate the data with priors, higher level models with a lot of overfitting, and poor identification of parameters, and yet it is not too noticeable. But if you do not engage in these practices, then it is essentially the same as maximum likelihood.
Except ML estimation gives you a point estimate, while Bayesian gives you a posterior distribution. Two different philosophies.

Bayesian methods help you to manipulate the data with priors, higher level models with a lot of overfitting, and poor identification of parameters, and yet it is not too noticeable. But if you do not engage in these practices, then it is essentially the same as maximum likelihood.
Except ML estimation gives you a point estimate, while Bayesian gives you a posterior distribution. Two different philosophies.
Yes, but only in appearance. The only reason the posteriori is interesting, is because it asymptotically converges to a degenerated distribution at the true parameter (a consistency result), so with large samples any draw from it is a valid estimator. But you need to choose one to play with your estimated model, so in the end you take the mode and that is asymptotically equivalent to ML. They are the same but using priors (that can move you to a bad place if you choose wrongly, and the sample is small).

Hierarchical bayes, once you substitute all the nested linear models for the beta parameters, end up as a standard linear model for the observed variables where you get very convoluted error structures, that introduce a lot of structure about the unobservable shock of the model, essentially a very complex random effect specification. This affects mostly to the efficiency, and if you are wrong you take the risk to get a very bad estimation. By contrast a fixedeffect OLS approach is simple and consistent (better from the Occam Razor perspective)

When theorists build models of rational agents, they always assume agents are fully Bayesian. Because Bayesianism is the only rational course of action.
When applied people do applied work, they are almost never Bayesian. I wonder why? Maybe they are not rational?Experimental evidence demonstrated the sure thing principle isn't really correct, so Bayesianism is on quite shaky ground especially for theorists, building off Savage. On the other hand, what you say is not correct there are nonBayesian approaches in both decision and game theory, although I admit they aren't well known.

Gents, what is the best book for learning Bayesian stats for someone who knows papa woolridge, mhe and some hayashi
Aka the regular econ grad student?
Whats the papa woolridge for Bayesians?There is a book by Ben Lambert of Youtube fame, A Student's Guide to Bayesian Statistics. Have not used it but does not look too technical, and Amazon reviews are good.

When theorists build models of rational agents, they always assume agents are fully Bayesian. Because Bayesianism is the only rational course of action.
When applied people do applied work, they are almost never Bayesian. I wonder why? Maybe they are not rational?there is some nonBayesian theoretical work but it is rare. There is also some bayesian applied work, but generally you are correct this is a huge divide between theory and applied.

When theorists build models of rational agents, they always assume agents are fully Bayesian. Because Bayesianism is the only rational course of action.
When applied people do applied work, they are almost never Bayesian. I wonder why? Maybe they are not rational?there is some nonBayesian theoretical work but it is rare. There is also some bayesian applied work, but generally you are correct this is a huge divide between theory and applied.
yep, see Binmore's Rational Decisions book for an attack on Bayesian Rational choice theory.
Savage argued bayesian rationality is for "small" worlds but the economy is not one of those worlds.