Bayesian statistics gives you a posterior distribution. What you do with that distribution is up to you. If you want to find the decision that minimizes the maximum loss instead of the expected loss, that fits perfectly well within the Bayesian framework. The posterior gives you all the information that you need to make a decision, whatever your loss function.

Frequentist statistics on the other hand gives you no such thing. You can't make decisions based on frequentist statistics. Frequentist statistics is all about reasoning about things that did not happen. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen. So while frequentist statistics is mathematically correct, it's also strictly speaking useless in practice. It's only useful insofar as it gives us heuristics for decision making when the Bayesian approach is impractical.

Here's an example. Suppose there are two types of berries: edible and poisonous. We devise a statistical procedure where you measure some properties of the berry (lets say we look at the color), and the procedure should help you decide whether to eat that berry. Now in frequentist statistics, you'll get a procedure that gives you the correct answer with probability at least p regardless of what your measurement was. Suppose that p=90% and we observe that the color is blue, and the procedure says: this berry is edible. Can we now eat the berry? No! This says absolutely nothing about the probability of the blue berry being poisonous or not. For example suppose 95% of berries are edible and red, and 5% of the berries are poisonous and blue. Then a valid procedure would be one that says that all berries are edible. It's correct >90% of the time, so yay! But if the berry we are holding in our hands is blue, it's incorrect 100% of the time. The fact that the procedure would have given us the right answer if the berry we found was red is irrelevant for making the decision of whether to eat the berry in the situation where the berry we found was blue. Things that did not happen are irrelevant for making decisions in a situation where that thing did happen!

tl;dr: frequentist vs bayesian is not about worst case vs average case. It's about P(measurement | true value) vs P(true value | measurement). The former is irrelevant for making decisions, the latter is exactly what you want.