> Much of mathematics is a formal affair of making arguments to satisfy necessary and sufficient conditions.

Some areas have a formalist culture like this (descriptive set theory seems to, for instance, though that might just be what it looks like from the outside), but it's far from universal. At the other end of the spectrum I find it hard to imagine anyone getting far with algebraic geometry without building intuition. And then of course in mathematical physics the intuition-frontier is always decades ahead of the formal one.

I would disagree. Mathematicians certainly don't need to visualize everything but "intuition" is a commonly used phrase which is a notion of understanding.

Although math merely requires proving some statement, often having an intuition / understanding of how concepts interact with each other helps figure out which things are likely to be true.

I never felt I properly understood a proof unless I understood it both intuitively and formally. The formalism is to make sure your intuitions are water-tight. But there are proofs you can accept are formally correct without intuitively understanding them - I would accept the truth of such proof but not feel like I understood them.