Indeed, by the time that you get to an introductory analysis course in the US (a 300-level undergraduate course!), students generally are confused about what the givens are in a proof, and what exactly they are trying to prove, let alone getting from point a to point b. Most have not had to "prove" anything since 9th grade geometry, and are used to plug-and-chug solutions backed by geometric concepts stripped of formality. Still, I think that few students would get to this point without having those concepts, and if I get confused on a problem my first step is usually still to "draw a picture"! The two go hand in hand in my opinion, but due to the demand from the engineering departments for teaching mathematical solutions (I don't know the full history of how the curriculum developed but I suspect this and its associated budgetary reasons are a large part), half of the learning process is severely delayed.