I don't mind the arbitrary cutoff point. That's like a Bayesian reporting likelihood ratios and leaving the prior up to the reader.

It's more things like, "And now we'll multiply all the significances together, and calculate the probability that their multiplicand would be equal to or lower than the result, given the null hypothesis" that make me want to scream. Why not take the arithmetic mean of the significances and calculate the probability of that instead, so long as we're pretending the actual result is part of an arbitrary class of results? It just seems horribly obvious that you just get further and further away from what the likelihood ratios are actually telling you, as you pile arbitrary test on arbitrary test...