The response paper seems to harp a lot on how the circumstance that typically, N_1 and N >> k, precludes any realistic applicability of the original theorem -- but it admits a fairly self-consistent and altogether much more sinister interpretation if you call the different \Phi "groups" (or "races" for maximum creepiness). The assumptions then say something to the effect of "every group's approach to problem solving produces fundamentally predictable results, and no group can be the best at solving each type of problem", which is a meme that in some form has been floating around in diversity activism circles for a long time.
The "mathematical theorem" then simply captures the triviality that under this sort of worldview, you want your team to contain Zorblaxians (sorry, SMBC) because there are some Zorblaxian problems that Zorblaxians have a natural affinity towards, and nobody without such an affinity could possibly make progress on. Since this is not stated explicitly, the statement becomes invokable even in settings in which otherwise a large number of people would raise eyebrows at the smell of exoticist quackery that the idea of "different ways of knowing" exudes.
Your post captures the problem much better than the linked article.
I would summarize the problem as being that the result is tautological, but the gloss of mathematics gives it the appearance of not being tautological.
If people said "diverse teams are better because people from different backgrounds are good at solving different kids of problems" then that would be accurate, but leave room for debate about whether then assumption is true. If they say "diverse teams are mathematically proven to be better", this would be inaccurate, but give great ammunition to argue that science supports progressive views.
The "mathematical theorem" then simply captures the triviality that under this sort of worldview, you want your team to contain Zorblaxians (sorry, SMBC) because there are some Zorblaxian problems that Zorblaxians have a natural affinity towards, and nobody without such an affinity could possibly make progress on. Since this is not stated explicitly, the statement becomes invokable even in settings in which otherwise a large number of people would raise eyebrows at the smell of exoticist quackery that the idea of "different ways of knowing" exudes.