I think intuition is hard to test in a way that feels 'fair'.

You can do it - I doubt you could have got a first when I was at Oxford just by learning and understanding the material, but you should probably have been able to get an upper second. The final part of every question virtually always involved insight, but you'd obviously then have to prove what that insight helped you understand.

If you give people questions like those, there is the risk of complaints about the university not having been taught the material for the exams I guess, or you might find that nobody can answer those harder intuition parts. Certainly most students at Oxford couldn't answer that many of them - you needed to answer about three 'final' parts out of about ten questions say in each three hour exam to get a first and perhaps about 20 percent of students got firsts?

Same here, but I didn't memorize the proofs, I tried to internalize their logic, so I could reconstruct them on demand by just thinking systematically. It did work for me pretty well on my real analysis final exam IIRC (27 years later).

Analogous to the Archimedean Property - there is no approach to teaching mathematics so intrinsically good that it cannot be done poorly enough to yield arbitrarily bad results.