I would still tend to think that the reference is a much bigger deal than the PS. No Oxbridge college is likely to admit more than a handful of Americans or more than one per subject. I’m pretty sure that my S18 was being compared against the other American interviewee at his college.
In that context if your teacher can’t say you are their best student in the subject in the last 5-10 years then (unless they say something like you are comparable to a prior student who was an IMO participant) you won’t be that impressive, given how relatively easy the APs and SATs are compared to A levels.
It’s a big difference to the UK where there is more of a track record of references. If my school said X is the best student in math or science, it was meaningful because they sent lots of kids to Oxbridge In that subject every year.
And the US applicants are being compared to kids from the Far East who are a) very impressive and b) likely to attend (because they have usually chosen to pursue either the UK or US systems but not both).
I do wonder if the high risk, high reward strategy in the future for math applicants will be to take STEP at the end of their junior year. That not only demonstrates skill, but also shows commitment and planning for attending a UK university.