Any school will offer plenty of different math classes, but this doesn’t mean they are designed primarily for students with different abilities.
Students enter with different goals (that is, they may be studying math, physical sciences, engineering, social sciences, etc). They might have different interests (how much do they enjoy abstract math?). Students also enter with varying levels of preparation (how advanced are they in math already?) and varying desires for acceleration and work load.
Since you mentioned Harvard, I looked through Harvard’s overview here https://www.math.harvard.edu/media/Math-course-information-for-first-year-students.pdf
I see plenty of tracks for different goals and levels of preparation. Courses are available for students who need math for various majors, as well as “students who have a particular interest in and enjoyment of abstract mathematics.” Students will also have different placement test scores, reflecting their preparation and comfort level with math. Finally, different math tracks are available depending on how much of a work load the student is looking for (“Math 25 differs from Math 22 by virtue of the work load in Mathematics 25 being significantly more than in Math 22”).
I particularly liked this quote:
The placement test does not distinguish between the courses in the Math 18, 19, 21, 22, 25, 55 and AM 22 column. What a student placed in that column takes depends partly on the student’s interests and partly on how much time the student wishes to spend on mathematics this year.
These are the specific courses that you mentioned, but it sounds like Harvard is classifying these based on the student’s goals, rather than ability.
Now, this is not to say that I don’t believe students HAVE differing levels of ability! Some students are just very good at math compared to others, and they may have an easier time in any of these math classes. But acknowledging individual differences in ability is a different thing than separating classes by ability. It may instead be sufficient to separate classes by student goals, student preparation, and desired workload / acceleration.