Let's assume our students are actual aspiring mathematicians.
Why would we introduce our students to Calculus rather than Real Analysis?
After all, "Calculus is a subset of Real Analysis". He will have to learn everything that he had already learned in Calculus once again. In addition to this, the student learns about mathematical concepts like axioms or proofs and mathematical working in general.
It's like saying: "We assume you're stupid, so learn Calculus first."
But I think this will make math harder instead of easier. For instance, If we just assume the real numbers as they are but do not show where they come from and how they can be constructed, the student will have a harder time understanding the key concepts of calculus.
Let's take the extreme case, where we just tell the student the differentiation rules. This might seem to be easier. But the student will ask himself what he's actually doing and why. Because of the missing theory, he can't grasp it. Because of that he will maybe see himself as a failure or even start hating math.
Well, historically we did not have any formalism. But our students aren't people from 2000 B.C.