I think the reason for Princeton poor ranking is being mostly an undergraduate school and hence not research oriented in terms of the amount of published scientific papers.
It is interesting because in most international rankings, which factor in research output quite heavily Princeton is in the top 10 usually. And yes while Princetin has a big undergraduate focus it does have many graduate programs that are top notch. maybe the fact that Princeton has no medical, law and business school could hurt it a bit in this ranking, but given the strength of the many graduate programs it does have I would still have expected it to rank top 30 if not top 20.
It’s not surprising about Princeton because the ranking includes quantity of publications as well as quality, and quantity of publications is going to be directly correlated to how large your graduate research programs are.
Princeton has only 2600 grad students, total. Looking at that list, I see that Harvard has 14,000 grad students, Stanford has 11,000, Johns Hopkins has 15,000, UWashington has 14,000, Berkeley has 10,000, UMichigan has 15,000, UPenn has 11,000 and so on.
I suspect that the correlation is pretty direct between number of grad students and number of publications, but I have no direct proof of this.