Regular school you would solve problems in combinatorics, algebra and geometry together in class by offering small ideas or strategies then trying each one to point out what works and doesn't work. You do enough of these everyday in class then reasoning about abstract sets in analysis isn't a huge leap. Same reason we used to read Euclid in class and make students demonstrate countless proofs.
CMU Discrete Math course has kind of the same techniques where he solves problems with the whole class https://m.youtube.com/watch?v=0K540qqyJJU
The best students then joined the school math circle with professional mathematicians and became TAs like in university helping their regular class.
They tried and failed the soviet curriculum with New Math https://simple.wikipedia.org/wiki/New_Math
Singapore high school students learn the equivalent of AP Calculus in Grade 10 then either spend years 11/12 at a trade school or stay to earn an associates degree to prep for university covering engineering electives like linear algebra and differential equations.