I'm not arguing which education system is better, but having a degree of familiarity in the subject matter you plan on proving things in helps out a lot. Proving things about continuity/discontinuity would be a bit difficult if you've never had to identify continuous functions with calculus.
Well, you're never going to deeply understand everything in a subject in only one or even several passes over the material, but there's no reason you can't concurrently treat analysis both formally and informally, with lots of concrete examples and intuition to go along with abstract definitions and proofs.
