The sibling gave some pointers. It's not that they're irrelevant. It's that they're historically important special cases, and decision theory provides the framework for thinking about when they make sense.

The general setup is this: you have a set of possible states of nature, a random variable on that set which will produce some observations, a set of decisions you may make, and a loss function defined over tuples of (state of nature, decision). The task is to produce a "good" function that maps from the observations to a decision.

It turns out that "good" isn't unique. You start by defining the expected loss for a given state of nature and decision function. Then you generally narrow your decision functions of interest to those for which there is no other function that has a lower expected loss for all states of nature (we call such ones "inadmissible" and ones that aren't uniformly dominated like this "admissible").

But then you end up with a whole collection of decision functions that are better under different conditions, and you need a way of choosing among them. Unbiasedness is one such criterion, as is maximum likelihood (though in many situations the maximum likelihood criterion may single out decision functions that are inadmissible), Bayes under a given prior, or minimax.

For some of these, particularly minimax, you also run into the fact that there aren't "enough" deterministic decision functions to select a best one...but if you allow randomized decision functions, there are.

Bayes procedures also enter in an interesting way: the class of Bayes procedures over all sensible priors is a superset of the set of admissible procedures...but usually not a much bigger set, so you can sometimes prove properties of all admissible procedures by proving them for all Bayes procedures.

For those who have studied game theory, this also leads to an immediate extension of repeated procedures where the decision rule at each step is "stop and accept a decision" or "sample again." These sometimes go under the name of "online trials" and are really useful. And since any admissible procedure is Bayes under some prior, you can often get away with doing the math for these online trials in a Bayesian framework, adding a hard coded stopping rule to correspond to your underlying decision procedure, even if you are a died in the wool frequentist.

The other, really important result in Bayesian statistics is that for a well behaved prior, data eventually drowns it out.

For my part, I don't take Bayesian statistics very seriously as a philosophical point of view. In the 1960's there was a lot of interest in it as a unified system of inference because there was a theorem showing that rational actors with a prior are Bayesian. When you start really digging into Bayesian statistics, though, you realize that a single prior is overspecified, and the more correct class of objects to work with is classes of priors. As soon as you do that, though, the rational actor is Bayesian theorem collapses.

I don't take the frequentist approach that seriously, either. Honestly, this is a space where I don't feel that there is a truly satisfactory underpinning besides decision theory plus "shut up and calculate." But that's true of so much of science that I'm not that bothered by it anymore.