Every time I read a rule about my mother tongue (Mandarin) online I was like, lol what nonsense foreigners made up... And then I realize that rule does exist. I just have internalized it for so long.
As a counterexample (in line with your explanation), consider someone snarking on the WallStreetBets forum: "Come on, guys, this is supposed to be Wall Street bets, not Wall Street prudent hedges!" Adjective order changes because the intended significance changes. (Normally it would be "prudent Wall Street hedges".)
Side note: please don't nitpick about whether "Wall Street" is functionally an adjective here. The same thing would happen if the forum had been named "FinancialBets".
It's a fool's errand because the way human language works is that people happily accept odd exceptions by rote memory. So the rule simply says that there exist these exceptions. Also, there is something called euphony: speakers find utterances questionable if they are not in some canonical form they are used to hearing. For instance "black & white" is preferred over "white & black".
The rules boil down to "what people are used to hearing, regardless of the underlying grammar offering other possibilities".
(I suppose Wall Street is a proper adjective, like "New York pizza," but you said no nitpicking)
By the way, modifying compounds generally must not be plurals, to the extent that even pluralia tantum words like scissors and pants get forced into a pseudo-singular form in order to serve as modifiers, giving us scissor lift and pant leg, which must not be scissors lift and pants leg.
An example of a noun phrase containing many modifying nouns is something like: law school entrance examination grading procedure workflow.
The order among modifying nouns is semantically critical and different from euphonic adjective order; examples in which modifying nouns are permuted, resulting in strange or nonsensical interpretations, or bad grammar, are not valid for demonstrating constraintsa mong the order of true adjectives which independently apply to their subject.
For instance, red, big house is strange and wants to be big, red house. The house is independently big and red.
This is not related to why entrance examination grading procedure cannot be changed to examination entrance grading procedure. The modifiers do not target the head, but each other. "entrance" applies to "examination", not to "procedure" or "grading".