There is no handwaving involved. The layman explanations of these phenomena are not that good. The quantitative derivations that leads to those results are totally fine, but hard to follow without deep knowledge of the subject.

Physics education sometimes aligns with historical evolution of the theories, mostly because that builds intuition and because the mathematical founsations of the improved theories need to be taught first. That leads to the "but in this case..." moments, but you need to realize that what you get taught as a "fix" is practically always a careful evolution that also reproduces all predictions from the less complete earlier theory.