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Most calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.

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Absolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function".

You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.

You're right. But then again, lots of scaffolding gets discarded when an arch gets constructed also.

Lest you disregard this as completely useless superstructure, note that basically the entirety of the theory of stochastic processes, starting with Brownian motion, is positively infested with continuous everywhere nondifferentiable functions; while the existence of a nonconstant infinitely smooth function with an identically zero Taylor series is what permits the Berezinskii-Kosterlitz-Thouless phase transition to exist. So while the weird animals of elementary real analysis are perhaps not the most important thing in the world, they are far from irrelevant to it.

It’s extremely niche (abstract, “Platonic”) to ever need to care about derivatives of non piecewise-analytic functions.

I think of wavelets and stochastic processes to be a significantly bigger niche than you probably do.

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