Not “most” high school students, but the more talented ones? Sure. Likely they were exposed to a lot of algebra before this lecture to bring them to a point where they could profitably follow along. The kind of kids who take AP Calc can learn abstract algebra (and, for that matter, I’d say that any kid who’s had Algebra II and properly taught geometry (with proofs) can learn abstract algebra).
The kinds of AP Calc kids you see must be super different from the ones I've seen then. I'm in a uni with a large amount of kids who've taken AP Calc and yet most of them struggled a lot with the first semester intro-to-proofs course. The second semester linear algebra course (it's a middle ground between a proof-based and a computational course) was even worse. I know many kids with 5's in AP Calc BC who resort to memorizing basic proofs (which is the sort of thing that helps with AP exams) instead of learning how to write one on their own, and some of the TA's have told me that the most common mistake on the midterm was incorrectly negating the statement "A is a subspace of B".
This is not to say that high schoolers can't do abstract algebra (or higher mathematics more generally). In senior year I attended a week-long camp (Arnold had a full semester) in my local uni where we proved the impossibility of squaring the circle, doubling the cube, etc using field extensions. And I was in the older side! Most of the kids there were 10th grades. Though Arnold's class was probably substantially harder than ours. I worked through V. B. Alekseev's book after the camp was over and the exercises were substantially harder than the ones we did at the camp. The material on Riemann surfaces was very hard to understand as well, much harder than the group theory part (I still don't understand Riemann surfaces lol).
In conclusion, AP Calc, and students' performance in it, is a terrible metric for assessing mathematical ability. Sorry for the long rant.
