One pitfall I'm wary of when introducing visual proofs is not being able to make the leap of how to formalize the proof, i.e. how to turn it into a purely mechanical process that a computer could understand.
It can make these sorts of proofs overly convincing. https://math.stackexchange.com/questions/743067/visually-dec.... My favorite is the approximation of the circle one, because it doesn't rely on tricky, underhanded drawing inaccuracies, but instead demonstrates a need to truly formalize what it is you're talking about.
For category theory, most people approaching it already have some experience with mathematical proofs and could probably sketch out how to boil a diagram chasing proof down into tedious set of logical statements. If anyone hasn't, I'd recommend doing so for a simple example.
Note a version of this can occur for the "algebraic" style of proofs as well. Occasionally students can't really explain why they're "allowed" to cancel out terms (it can be a minor leap to see that really what's being relied on here is injectivity).
The other tricky thing about intuitions, visual or otherwise, at least in my experience, is that I often hold multiple mutually incompatible visualizations/intuitions about a mathematical object or process and the most crucial component of my intuition is knowing when to discard one and use the other when they conflict. To actually harmonize all of them requires, well, fully formalizing everything. Otherwise you end up mistaking your intuition for the object itself and going down a logically incoherent path (the evergreen target for this always seems to be Godel's incompleteness theorems).
You still need intuition though, because otherwise coming up with the creative spark for a proof is nigh impossible. But it's not a substitute for the formal object itself.
More fundamentally, I think both approaches, visual and "algebraic" in the sense of the article make it seem like mathematics is about getting the "correct" answer, when really the part of pure mathematics that resonates most with me is about running wild with "what if" and then rigorously chasing down the implications thereof.
For example, the commonly asked playground question "is infinity number?" is not best answered with a "no" or a "yes", but rather an exploration of what no and yes would entail, which first requires the formalization of infinity, which could have many different, mutually incompatible forms! Another fun one is coming up with a world where infinity plus one is larger than infinity (this often leads to an exploration of the ordinals).
I also enjoy how relevant your username is.