And if you expected to reject the null hypothesis, isn't that failure meaningful?

To me, but not to the theoretical foundations of the method employed.

Hypothesis testing generally works sensibly because people smuggle in intuitions that aren't part of the foundations of the method. But since they're only smuggling things in under a deficient theoretical framework, they're given to mistakes, particularly when they're applying their intuitions to the theoretical framework and not the base data.

I agree with the later comment on Bayesian statistics, and I'd go further. Scatterplot the labeled data, or show the distribution if you have tons of data. That's generally much more productive than any particular particular confidence interval you might construct.

It would be an interesting study generative study to compare the various statistical tests on the same hypothesis versus the human eyeball. I think the eyeball will hold it's own.