The reason you should study calculus isn't to learn to integrate and differentiate; it's to overlearn basic algebra. If you are designing a proposed statistics-based calculus-replacer, it needs to give extensive experience with basic algebra.
My memory of statistics classes is fuzzy, but my impression was that it didn't give as many opportunities to practice algebra - algebra did occurr in proofs, but the proofs were beyond the freshman/sophomore level, and not routinely assigned as homework.
That's an intriguing idea, and now that you mention it, there are other classes that seem to have similar purposes. Physics gives you practice with basic calculus, and electromagnetics involves more multivariate calculus than you can shake a stick at.
This guy says that the problem is that high-school math education is structured to prepare people to learn calculus in their freshman year of college. But only a small minority of students ever takes calculus, and an even smaller minority ever uses it. And not many people ever make much use of pre-calc subjects like algebra, trig, or analytic geometry.
Instead, high-school math should be structured to prepare people to learn statistics. Probability and basic statistics, he argues, are not only more generally useful than calculus, they are also more fun.
I have to agree with him. What do the people here think?