I sympathise with your point of view and I agree that sometimes mathematicians are not keeping their audience in mind enough. You don't always need completely rigorous proofs for non-mathematicians.

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.