“SAT scores become relatively less important the higher ranked the school is”
Not really. As a high stats kid, your scores may not distinguish you among so many other high stats kids, work any special magic. But scores can still be a sort of benchmark. Harvard isn’t giving a pass to a lower scoring kid, just because they have so many high stats applicants.
Nor are they de-emphasizing scores because they come in 3 varieties. Each tippy top knows how to view the New SAT in relation to the old. I presume the same goes for other tiers.
OP, as others say, the std tests may be the same, across the country, but results need to be interpreted in the high school and individual context.