No probably needs a couple more years. Writing the test itself is motivation to do well in HS math. If that no longer becomes a driver probably takes off the drive in other courses over a couple years. I bet without the SAT as a standardized test a lot of HS math courses are easier for the teacher because the quality can lapse.

Also some children who excel write their SATs sometimes 2-3 years before college and then re-write if need be.

Not if kids are prepping for the test in a way that results in real gains. Which seems likely, especially in the age of AI: "should I actually study math or just use ChatGPT to pass this course?" One semester of coasting through might not do that much harm, but at some point the compounding effects will tip you over the edge.

Define "right away". How long after taking the SAT does a child have their first classes at university? At least a year?