In calculus the core issue is that the concept of a "function" was undefined but generally understood to be something like what we'd call today an "expression" in a programming language. So, for example, "x^2 + 1" was widely agreed to be a function, but "if x < 0 then x else 0" was controversial. What's nice about the "function as expression" idea is that generally speaking these functions are continuous, analytic [1], etc and the set of such functions is closed under differentiation and integration [2]. There's a good chance that if you took AP Calculus you basically learned this definition.

The formal definition of "function" is totally different! This is typically a big confusion in Calculus 2 or 3! Today, a function is defined as literally any input→output mapping, and the "rule" by which this mapping is defined is irrelevant. This definition is much worse for basic calculus—most mappings are not continuous or differentiable. But it has benefits for more advanced calculus; the initial application was Fourier series. And it is generally much easier to formalize because it is "canonical" in a certain sense, it doesn't depend on questions like "which exact expressions are allowed".

This is exactly what the article is complaining about. The non-rigorous intuition preferred for basic calculus and the non-rigorous intuition required for more advanced calculus are different. If you formalize, you'll end up with one rigorous definition, which necessarily will have to incorporate a lot of complexity required for advanced calculus but confusing to beginners.

Programming languages are like this too. Compare C and Python. Some things must be written in C, but most things can be more easily written in Python. If the whole development must be one language, the more basic code will suffer. In programming we fix this by developing software as assemblages of different programs written in different languages, but mechanisms for this kind of modularity in formal systems are still under-studied and, today, come with significant untrusted pieces or annoying boilerplate, so this solution isn't yet available.

[1] Later it was discovered that in fact this set isn't analytic, but that wasn't known for a long time.

[2] I am being imprecise; integrating and solving various differential equations often yields functions that are nice but aren't defined by combinations of named functions. The solution at the time was to name these new discovered functions.