Let me see if I understand your claim:

I would only qualify my earlier statement: while human intelligence is flexible enough to understand anything that is possible, it might not be large enough. If there's too much going on, the brain may simply not be able to compute it. In which case, the non-understanding doesn't feel non-intuitive, it just feels too complicated.

I question whether the intuition always comes in the form of a mechanism, or even any additional concepts at all.

Even correct intuition? I guess I don't mind putting forth a more definitive assertion that intuition must be based on a mechanical understanding. (While it's likely I'm wildly guilty of the typical mind fallacy, that's nevertheless my view.)

I've been considering the hypothesis that mathematical intuition (especially intuition about highly abstract, non-physical things) comes from an ability to model that math physically in the brain. When we interrogate our 'intuition', we're actually interrogating these (mechanical) models. Modeling is a high-intelligence activity, and a model 'correct enough' to yield intuition may be hardly recognizable as such, if we were forced to explain in detail how we knew.

(If we have a correct intuition about mathematics outside our experience, how else could we have it?)