This is going to sound silly, but...could someone explain frequentist statistics to me?

Here's my current understanding of how it works:

We've got some hypothesis H, whose truth or falsity we'd like to determine. So we go out and gather some evidence E. But now, instead of trying to quantify our degree of belief in H (given E) as a conditional probability estimate using Bayes' Theorem (which would require us to know P(H), P(E|H), and P(E|~H)), what we do is simply calculate P(E|~H) (techniques for doing this being of course the principal concern of statistics texts), and then place H into one of two bins depending on whether P(E|~H) is below some threshold number ("p-value") that somebody decided was "low": if P(E|~H) is below that number, we put H into the "accepted" bin (or, as they say, we reject the null hypothesis ~H); otherwise, we put H into the "not accepted" bin (that is, we fail to reject ~H).

Now, if that is a fair summary, then this big controversy between frequentists and Bayesians must mean that there is a sizable collection of people who think that the above procedure is a better way of obtaining knowledge than performing Bayesian updates. But for the life of me, I can't see how anyone could possibly think that. I mean, not only is the "p-value" threshold arbitrary, not only are we depriving ourselves of valuable information by "accepting" or "not accepting" a hypothesis rather than quantifying our certainty level, but...what about P(E|H)?? (Not to mention P(H).) To me, it seems blatantly obvious that an epistemology (and that's what it is) like the above is a recipe for disaster -- specifically in the form of accumulated errors over time.

I know that statisticians are intelligent people, so this has to be a strawman or something. Or at least, there must be some decent-sounding arguments that I haven't heard -- and surely there are some frequentist contrarians reading this who know what those arguments are. So, in the spirit of Alicorn's "Deontology for Cosequentialists" or ciphergoth's survey of the anti-cryonics position, I'd like to suggest a "Frequentism for Bayesians" post -- or perhaps just a "Frequentism for Dummies", if that's what I'm being here.