You misinterpret the law of large numbers. What the law says is that if you have a large amount of samples, and assuming there's no pervasive bias in the samples, then any large enough sample (and often that's much smaller than you think - the classic example being election voters, with a group of only a few thousand representative voters being enough to predict the outcome of an election over a large country with millions of voters) will look identical to any other... that is, over a large enough sample, in the case of this article, the conclusion of many papers should converge to the same answer, with outliers being marked out as likely "bad" papers.

The only assumption you may reject here is that there's no systematic bias in the papers. Perhaps there is... or perhaps most papers are just very unreliable, in which case there should also be no convergence... but if you find convergence, there's a good chance the result is "real".