In 1965 I had the idea that the proof of the Seifert-van Kampen theorem for the fundamental groupoid generalised to two dimensions, and higher, but lacked the gadget corresponding a 2-dimensional fudamental groupoid using squares composed in two directions. So this was an idea of a proof in search of a theorem. So I tried for 9 years to define this for a topological space. Finally, in 1974, Philip Higgins and I realised that we could do this for a pair of spaces, i.e. a space $X$ and subspace $A$, mapping a square to $X$ with edges mapped into $A$ and taking homotopy classes of these maps with vertices fixed in the homotopies. Fortunately, lots of work on related algebra had been done in the meanwhile, so the main stuff rolled out, and got published in 1978.
Unfortunately, the use of groupoids and double groupoids seemed to arouse hostility. so this and the work in all dimensions was, a colleaue remarked, pursued in the teeth of opposition!
So that is another possible affect of intuition. to say some work is ridiculous! It's a hard life! But has been lots of fun pursuing a line of intuition and trying to make it really work. I was lucky in my collaborators, too.
Later: There is an example in J.E. Littlewood's "A mathematican's miscellany" where a picture contains the essential argument. In higher category theory, there is quite a lot of use of manipulating diagrams, and this is regarded, rightly, as rigorous.