It’s not that. Consider the definition of the limit. The idea existed for a long time. Newton/Leibniz had the idea.

That idea wasn’t formally defined until 134 years later with epsilon-delta by Cauchy. That it was accepted. (I know that there were an earlier proofs)

There’s even arguments that the limit existed before newton and lebnitz with Archimedes' Limits to Value of Pi.

Cauchy’s deep understanding of limits also led to the creation of complex function theory.

These forms of creation are hand-wavy not because they are wrong. They are hand wavy because they leverage a deep level of ‘creative-intuition’ in a subject.

An intuition that a later reader may not have and will want to formalize to deepen their own understanding of the topic often leading to deeper understanding and new maths.

> Newton and Leibniz were "hand-waving"?

Yes, and it's pretty common knowledge that Calculus was (finally) formalized by Weierstrass in the early 19th century, having spent almost two centuries in mathematical limbo. Calculus was intuitive, solved a great class of problems, but its roots were very much (ironically) vibes-based.

This isn't unique to Newton or Leibniz, Euler did all kinds of "illegal" things (like playing with divergent series, treating differentials as actual quantities, etc.) which worked out and solved problems, but were also not formalized until much later.