On a factual level the relationship between kinship societies and economic headwinds is fairly well documented [1] [2]. The mechanism is the same reason that communist/socialist societies often fail: when wealth belongs to everyone, nobody has either the incentive or the means to accumulate wealth, which prevents capital formation within the society [3].
The part that the article glosses over is that "Kinship societies destroy economic growth" is a Russell conjugate [4] of "economic growth destroys family formation". Kinship networks provide important intangible support to several important community functions, notably child-rearing. That's the whole "it takes a village to raise a child" aphorism. When you allow people to defect on their social obligations in the name of accumulating wealth, then it turns out they do, and the village suffers. It is exactly as the article said: "The kinship network has a strong interest in preventing any of its members from becoming prosperous enough to no longer need it: someone who no longer needs your help is also someone who might not help you." That's exactly what we've observed happening in modern industrialized economies, where people become increasingly atomized and those informal community organizations that create things like belonging and mutual aid (not to mention group childcare and socialization) die off as everyone chases the promotion that will let them afford ever-higher institutional childcare costs.
And this is why the fertility rate in every major industrialized country has cratered, usually right as it industrializes.
[1] https://www.uni-heidelberg.de/md/awi/forschung/paper_e.bulte...
[2] https://edepot.wur.nl/14918
[3] https://en.wikipedia.org/wiki/Tragedy_of_the_commons
[4] https://en.wikipedia.org/wiki/Emotive_conjugation