The world's leading science journal publishing grand claims without verification, in the face of mounting evidence that the claims might be incorrect, is most certainly not "the scientific method".

> It would be entirely infeasible for them to require someone else to test the sample.

What nonsense. Why would that be infeasible?

But seriously, why can't they ship their sample somewhere?

Peer review serves basically as a smell test where capable experts certify that the paper is novel, interesting, properly justifies its claims, and contains enough detail to reproduce the results. It is not uncommon for only a few labs to contain all of the expensive, specialized equipment to reproduce an experiment, and getting that experiment working (if it is reproducible) can take a significant time commitment from multiple individuals in a lab. Also keep in mind that reviewers are normally not paid for their services, journals typically want at least two independent reviewers (who they pick, sometimes with input from the author to narrow down who has the background to perform the review), and going from submission to publication is typically a many-month process without the high standard of reproduction.

Reproduction happens in the manner that is going on right now. If your publication makes waves, people will want to build on your result and either reproduce it as a first step or conduct a related experiment that adds evidence to support the hypothesis. This way is far more practical, but the devious thing is that, if you lie and manipulate your data, some poor grad student might waste two years of their PhD trying to get your thing to work.

Having journals or some independent body fund reproduction themselves runs into both capital challenges and the problem of attracting a large enough number of capable scientists to cover every discipline who would only want to work on reproduction full time. As nice as that would be, I’m not sure such a venture would work in the real world.