I would put a good deal of the blame for this squarely on frequentism as well. Bayesianism isn't hard to understand, it's just takes an effort of the teacher to explain well - I've made certain notable efforts in that direction myself. Once you do get it, you get it.
Insert results, get a green or red "significance" light, move on.
A great article on the problem with statistical significance (by a Frequentist) is here, called Why Most Published Research Findings Are False: http://www.plosmedicine.org/article/info:doi/10.1371/journal...
The punchline seems to be, well, that there's also a large human element that contributes to the problem. I think it's one thing to rail on the Frequentist way-of-thinking; it's entirely another to state that the institution of scientific research built on it creates unwanted incentives.
Often (always?) these constraints are all so very much more complex than Gaussian power analysis states. You do it as a way of sketching the depth of a problem I think, not much more.
The linked paper is a pretty clear introduction of the high level problems. I think it's perhaps a little more grim than necessary, but then again that might just be my own bias.
If you try to press on what people believe "significant" to mean it gets ugly fast, but it's generally a good thing and definitely necessary to publish. If you don't get significance it's just because you need a bigger n. If you can think of some factors or covariates then you really need to use ANOVA.
Stating that there is anything more complex to looking at data and deciding what it means is practically unthinkable.
Likelihood ratios are definitely nice, though. I've sort of gotten people to think about it at a high level by talking about "information flows" and log likelihood values.