Any high school administering the SAT or any other standardized tests knows what they need to do. They know who has accomodations and who doesn’t. I don’t see a need to assume they don’t know how to do their job.
FWIW, my son had extended time. I sent him on his way on test day, I didn’t contact the school he tested at, I didn’t send him with anything other than his test day ticket, and all was well. It is uncommon for CB to get it wrong, honestly.
I see no reason for anyone to take extra steps to be sure their child gets what College Board has approved well in advance. All the extra measures are taken long before a kid sets foot in a testing center. A student can’t get extended time without prior approval from CB, typically via the school, with a whole lot of supporting documentation, submitted by the school.
If the student referred to by the OP has testing accommodations, everyone at the schools will know because everything is planned out well in advance. In fact, when signing up for a test site, a student with accommodations may not even get into their first choice test center if the test site hasn’t got enough space set up for all the students with extended test time.