No, that's my bad, thank you for correcting me! I only read the abstract, and they don't mention Bayesian methods there. Confidence intervals suffer from similar flaws as p-values and statistical significance.

It's great that they do analyses with other methods too indeed. Not, from my point of view, because they're more approachable – quite the opposite: people think in terms of probabilities-of-the-hypotheses, and p-values are not that (that's one source of their misuse). But because it helps the transition to other methods. It'd been nice if they had stated the results from all methods in the abstract. But that'll be for next time maybe!

P-values-based methods and statistical significance are flawed: even when used correctly (e.g.: stopping rule decided beforehand, various "corrections" of all kinds for number of datapoints, non-gaussianity, and so on), one can get results that are "statistically non-significant" but clearly significant in all common-sense meanings of this word; and vice-versa. There's a constant literature – with mathematical and logical proofs – dating back from the 1940s pointing out the in-principle flaws of "statistical significance" and null-hypothesis testing. The editorial from the American Statistical Association gives an extensive list.

I'd like to add: I'm saying this not because I read it somewhere (I don't like unscientific "my football team is better than yours"-like discussions), but because I personally sat down and patiently went through the proofs and counterexamples, and the (almost non-existing) counter-proofs. That's what made me change methodology. This is something that many researchers using "statistical significance" have not done.