I agree completely, but calculus touches so much of advanced math that at some point you have to learn it in depth (assuming you actually want to learn advanced mathematics). I'd argue that linear algebra is a more interesting subject for most people, but calculus contains more important ideas that are crucial if you intend to continue studying mathematics.
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I have a major in Mathematics, and I truly didn't need calculus after the multivariable/vector calculus course I had to take my Freshman fall. Why did I have to do so much if it wasn't going to be necessary until graduate school at the earliest, or perhaps never depending on specialization? And why was it a prerequisite for my other math courses (in algebra, topology, real analysis, number theory, and combinatorics)?
How was topology motivated without talking about generalizations of limits?
(Honest question, no snark.)
It's a good question.
I think my first topology course, like in many intro analysis courses, shared concepts from calculus without requiring any previous expertise from our calculus courses. A rigorous approach to real analysis or point-set topology will be very set-theoretic, and you build the general definitions of continuity, compactness, etc from the ground up. So, I would have used concepts from my analysis and topology courses in calculus, if only it had been taught in the other order.
I guess it feels like limits are a rigorous foundation for calculus, but they are not calculus. In any case, I didn't need anything from differential or integral calculus (or that entire required course developing the theoretical basis for Stokes...), which is what I mean to say.