Or maybe even just write some mathematical overviews myself that leave out stuff that drives mathematicians insane with frustration...
It would be justice...
But I think that it's a false dichotomy to only consider "extremely rigorous" or "simplified to the point of being wrong" as the only teaching philosophies. There is a middle ground where you can be somewhat hand-wavy and inexact without getting the conceptual broad strokes wrong.
No, but you might tolerate example code with mild syntax errors (like using a "..." to skip unrelated material, or skipping import declarations and boilerplate to better clarify the point being made). You don't demand that every example in a programming tutorial be a fully buildable piece of software, why are you demanding that a simpified calculus tutorial exhibit full rigor?
Aren't they just suggesting a marginal amount of rigor to communicate that the "definition" is in fact based on justifications which render it non-arbitrary?