That's a very insightful way of looking at it. If Nature wanted to get out of this unfortunate position, it would have to counteract this with stricter requirements on the articles that it publishes. It could require that papers be replicated before publication; it could require the release of any relevant code and data; it could be really diligent about publishing failure-to-reproduce papers. This would be a hassle, but would probably make Nature more prestigious, so I'm sure they could get a lot of researchers to go along with whatever requirements they want.
Nature's and Science's shenanigans are not publishing papers that turn out to be wrong, it's that (1) when contrary data comes up they refuse to publish it, as in this case, and (2) really questionable editorial policies, where articles get peer-reviewed by the wrong audience, and on-the-face obviously-wrong to anybody in the field papers still get published.
Every journal publishes novel and statistically significant findings, and probably encounter similar rates of type 1 error as Nature and Science. It's just that Nature and Science are more susceptible to human politicking and don't seem to face up to it.
> This would be a hassle, but would probably make Nature more prestigious...
Prestige is a relative thing, and there's nothing more prestigious than Nature (or Science). Your suggestions are good, but I think they would be more likely to be adopted first by a journal that isn't Nature or Science, that possibly wanted to compete with them.
