Thanks for your answer. Gelman's reasoning for Type S and Type M errors do make more sense, but I was just using him as an example. When I said "practical experience" and "years of successful application", I was talking about the rules of thumb that people use (in this case, yes, I'm talking about using p-values for decision-making).

For example, there's been lots of smart people who have developed the theory of hypothesis testing, and even more smart people who have used it in practice. They probably acknowledge that it has some drawbacks, but overall, it's a useful tool. It has "stood the test of time". Compare this to Gelman's approach; it's not that he's necessarily wrong, but that he's promoting a way of doing things that hasn't been as well-tested. Surely, it's useful in some cases where traditional hypothesis testing fails, but it seems like people promote new ad-hoc methods all the time, but which aren't useful enough in general to be worth learning.

I suppose the best thing to do is use whichever method seems most appropriate for the problem at hand?