>I think this is more in the lines of mixing up concepts (the language features that enable implementing functors etc. in Haskell with functors as used in programming).

You know how you can identify an fp person? They'll condescend to you and dismiss what you say.

>Type classes (as a language construct) are a way of doing ad-hoc polymorphism (read run-time operator overloading)

You're glossing over what I'm saying and drawing some superficial analogy that I am not - I'm not talking parametric polymorphism, I'm talking about monadic evaluation ie evaluation within a context. I'm talking about the same thing as dynamic binding, which is also used to implement the same exact things as monads (see the collapsing interpreters paper by Amin and Rompf).

So no I'm not confused and I'm not conflating, I am in fact making a strong claim. Again, my proof is exactly using the same language: a morphism between runtime operator overloading and lift/bind.

To give a more abstruse argument, compare Conal Elliot's compiling to categories and jax (which is immediately recognized as a monadic interpreter framework).