Well, the most commonly used statistical methods are probably:
All of these are frequentist: logistic regression is quite explicitly computing the maximum-likelihood-estimate of a parameter vector, SVMs are minimizing a surrogate to generalization error, and PCA is a bit weird but is basically just trying to find a low-rank approximation to the data.
ETA: And to answer your other question, I think that would just be called "the best method"; why would we need another name? No one is going to design a method that they think is strictly dominated by all other methods anyway...at least, not if you also take into account time to implement, which I think is an important consideration (at least in the limit, where with infinite time I can just hard-code everything as a special case).
ETA2: It's also not clear to me that Bayesianism is correct in the ideal sense (or even what that means), or that it's fruitful to think of what you're doing as trying to approximate Bayes (at least not in all situations; I definitely agree that it can be helpful sometimes). I don't know if I'll be able to convince you of either of these here though, as this is a disagreement that Eliezer and I still have despite a 4-hour-long discussion (and of course this causes me to update in the direction of me being mistaken).
It's also not clear to me that Bayesianism is correct in the ideal sense (or even what that means)
Interesting. Do you accept that by Cox's theorems, probability theory is the normative theory of epistemology? Do you accept that a "bayesian" method based on explicitly approximating ideal probability theory will always give a more accurate answer? Do you accept that each of the examples above work because and to the extent that they (nonexplicitly) approximate the correct probability-theory answer (the bayes-structure argument)?
(as for how they...
Question in title.
This is obviously subjective, but I figure there ought to be some "go-to" paper. Maybe I've even seen it once, but can't find it now and I don't know if there's anything better.
Links to multiple papers with different focus would be welcome. For my current purpose I have a preference for one that aims low and isn't too long.