I should preface these questions about the Italian school of algebraic geometry by stating that I'm asking for the sake of a short story. This means two things for the sorts of answers that would be most helpful to me. First, historicity is of course preferred, but apocrypha and rumor and conjecture are perfectly fine if they communicate something interesting. I'd be taking creative liberties in any case. Second, although specific mathematical examples are welcome and likely necessary to illustrate the point (e.g. how they might have thought about generic points of a variety), I'm most interested in philosophical or even psychological takeaways.
Would the Italian algebraic geometers have been concerned with the philosophical notion of rigor? I imagine a meaningful difference between believing one's methods satisfy some (ill-formed) notion of rigor and believing one's lack of rigor is irrelevant to an answer being satisfactory. So I'm curious what side they would have been on.
Do we have examples of mathematicians working in this intuition-led approach whose careers overlap the Zariski/Weil and even Grothendieck-style formalism? If so, what do we observe? How do they transition or react, if they do?
There are certainly no "correct" answers to these questions – anyone's opinion or interpretation is valid and interesting to me.