"Why doesn't the preeminence of the US math knowledge appear to seep into the primary and secondary school education?"
Because preeminence of top tier institutions (which are kind of global centres anyhow) - has absolutely nothing to do with teaching math to the commons.
Here's a hint:
+++ Americans don't suck at Math +++
There's a very un-PC but very large elephant in the room that people won't discuss.
+ European American and Asian American 'testing scores' are actually pretty good - and have been holding steady for a very long time. (Asians do a little better). Nothing has changed.
+ Latino American and African Americans fare poorly, but having been getting better since we've been measuring by standards (i.e. 1950's-1970's).
Here's the trick:
+ European Americans actually do better than Europeans - on average.
+ Asian Americans to better than Asians - on average.
+ Latino Americans do better than Central/South American Latinos
+ African Americans do better than Africans.
The key correlating factor here is 'ethnicity'. 'Ethnicity' is the broad, generalist predictor of educational outcomes. This definitely not 'race' and it's not even 'IQ' (those things are plausible but controversial) - it's a series of behaviours, social norms, examples, attitudes towards work, success, access to services, social networks, mentors, role models, etc. etc. etc..
Educational outcomes (and crime stats, income stats) break down along ethnic lines. In a manner of speaking - America can be thought of as 'four nations' - White, Black, Asian and Latino. Obviously - it's very crude and generalist, and policy based on this would probably be racist - nevertheless - you pretty much have to look at the data given this.
In the end: American test score results have more to do with the changing ethnic composition of the American population than they do anything else. Again: White people and Asians in America have performed consistently he same for decades. Teaching methods haven't changed much, students habits haven't changed much - so the outcome is naturally consistent.
More economic prosperity, access to services and different attitudes + deeper integration have meant Latino A. and African A.'s are doing a little bit better - but because there are so many more Americans of those groups - particularly Latino Americans - it changes the outcome of the 'average american test score'.
Analyzing educational results does not make sense until you break it down along ethnic lines. Once you do - it becomes crystal clear. It's the absolute #1 most important thing about the educational data that turns 'paradox' about educational investment (teaching has remained largely the same) and outcomes into 'perfect sense'.
Unfortunately, it's so sensitive few will want to talk about it - for fear that the general public equates educational outcomes to 'intelligence' and try to strongly correlate ethnicity + race to this, which would be fodder for racist/KKK types, which wouldn't really help the overall social situation in America.
Anyhow - America is actually doing pretty well overall.