I'm mostly on board with the critique of university rankings, but I really don't buy this specific critique. It's a strawman.
Here's what they did: they took the top research universities and the top liberal arts colleges. They then observed that the rankings can't distinguish between the top institutions within each class, and that you can also flip which of those two classes is preferred.
But, in what sense is it even meaningful to compare Amherst College and Harvard [1]? Those are just enormously different types of institutions. It'd be like creating a ranking of "best cars" that includes the top 5 sedans and the top 5 trucks, and then observing that you can jiggle the rankings to get anyone on top. Is a Toyota Corolla better than an F-150? IDK. Stupid question. US News and World Report, for all its problems, does at least get this much right! they break down institutions by "type" and then rank within type.
Additionally, just because rankings are noisy and easy to game locally doesn't mean they are inaccurate or easy to game globally. Two institutions within 10-20 slots of one another are probably pretty similar and rankings aren't particularly helpful / are easy to game. But the #100 liberal arts college is probably not as good as Amherst, and the #78 National University is probably not as good as Harvard, and no amount of gaming is going to change that.
Rankings are indeed noisy and inaccurate and easy to game. But this particular article is not a compelling demonstration of that fact.
[1] For non-US readers: Amherst college (not to be confused with UMass Amherst) is in a class of peculiar institutions that are fairly unique to the USA as described here: https://en.wikipedia.org/wiki/Liberal_arts_college TL;DR: they're basically the diametric opposite of super-charged research universities like Harvard.