The SAT is treated as a standardized test but it, as you note, is inaccessible to many. It is not government run or organized. I've found, from teaching undergraduates, that most of the students who go to college presume that everyone takes the SATs, that it is government administered, and that it is free.
compare this to the, still far from perfect and problematic in other ways, centralized university admissions testing system in many other countries.