Let me throw a bit more complexity into the mix. I graduated from high school in New York City almost 40 years ago. My kids went to school in New Jersey and then California.
When I went, elementary school was K - 6, kindergarten being a half day. Junior High was 7 - 9, and High School was 10 - 12, though they moved 9th grade school about the time I graduated. To add complexity, “honors” students (called SP, or special progress ) had an option of going through junior high in 2 years, and about half of us did that. (Not me.) My high school was very big, about 1500 in my graduating class, and had Honors, Extra Honors, and AP classes. There were a lot fewer AP classes back then than there are now. What HippyHollow said may hold for Texas, but no where I’ve been equates honors students with students going to college. While almost all honors students go to college, lots of non-honors students do also.
In NJ and California we have the traditional K -6, 7-8, and 9 -12 breakdown, though in NJ our district was going to a K-5, 6-8 and 9-12 format. In addition, the NJ district had a “pre-first” program to delay the move from Kindergarten to first grade for a year. This was used for students who were young. We held our daughter, who would have been the youngest back from going to Kindergarten for a year, which worked well. We knew people who put their kids in pre-first, and it worked out great for them.
Many states have exams that you have to pass to graduate now. California allows several tries. They aren’t stressful except for marginal students, as the Regents exams (which took in NY) weren’t stressful either.
As for exams, many seem to be linked to scholarships. There are National Merit exams, and New York had a Regents Scholarship exam which gave you money if you did well enough, but only if you went to college in New York. Most kids take periodic exams for purposes of measuring school progress and for No Child Left Behind.
As for colleges, no private schools are anywhere near open admissions. Most community colleges are. Most states with open admissions have it only for lower tier state schools - the University of California schools like UCLA and Berkeley are by no means open admissions. Even the CSUs require a certain grade level to get in.