I think that I just take issue with the term "hand-waving" as equated to intuition. Yeah it lacked formal rigor, but they had a solid model that applied in detail to the real world. That doesn't come from just saying, "oh well, it'll work itself out". I guess if you want to call that "hand-wavy" we'll just have to disagree.
Euclid disproves every bullshit posted by LL Mediocres unable to understand that before Calculus there were proto-calculus based ideas such as Zeno's paradoxes and some writtings from Archimede which pretty much are Calculus 0.9.
Americans and British geeks/nerds are blinded down by Newton unable to realize that there was tons of previous work since the Greek and in Middle Ages, where the British love to depict as brutish people with no culture at all.
And the case is that they weren't dumb at all and without Euclid and Archimede there woudn't be any Calculus.
https://en.wikipedia.org/wiki/Euclid%27s_Elements
https://en.wikipedia.org/wiki/Method_of_exhaustion