None of these examples contradict what I've said. The historical fact is that continuous functions and vector addition were discovered as interesting mathematical properties based on certain visual intuitions. Only later were did they come to be extended to more general notions applicable to wildly different conditions.

What I claim in addition is that it's still useful for your intuition to understand this history and the leaps that were made by extremely talented mathematicians of the past who came up with these formalizations of intuitive properties.

I'd also claim that they couldn't have ever arrived at the current formal systems if they hadn't started with certain intuitions for simple systems.