I literally don't understand what the hell you are talking about.

But this has nothing to do with constructivism. Even if you only allow constructive definitions and proofs, there is still the world of a difference between the definition of an integral and the result you get from evaluating it.

Yeah, sure, in theory you can represent an integral as a function that takes another function and two boundary points and returns a value...

But first, it may not be possible to determine the value of the integral exactly because there is no known method of doing so (the Risch algorithm, apart from it basically being so complex that it's implemented almost nowhere fully, only works for elementary functions!).

And second, if integrals are "just functions", you lose the ability to manipulate them according to known theorems, e.g. additivity, triangle inequality, Cauchy-Schwarz, convergence theorems, ...

So yeah, here's where I get the feeling that some people should do some more maths and spend good parts of their days proving theorems and playing with definitions before they start complaining about how dumb its language is.

If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. Even then, I think your claim that every integrable function has a "generalized antiderivative" is also only true for the Riemann integral. The Dirichlet function is Lebesgue integrable, but it doesn't have an antiderivative even in this weaker sense.[^1] And mathematicians generally prefer the Lebesgue integral.

I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition.

And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all.

[^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though.