I think that there are several reasons:

1. Mathematics require development. Just like the switch from two oranges + three oranges, to a more abstract idea of 2+3 requires time, so is the ability to think clearly about abstract spaces, infinities, functions and relations. It takes time and experience.

2. Mathematics is like a muscle. You need to work out a lot to have a big muscular brain that can bench-press theorems, later on to be able and maybe invent some of your own (i.e. research)

3. When most people come to the university they don't know what math is like really. Before my freshman year I wanted to study pretty much every course that would be given. After a while I realized that analysis is not my cup of tea, while set theory is. And although algebra seems nice at first, eventually it goes beyond the scope of my liking. Today as I'm taking my first steps into learning on my own and starting my M.Sc. I can tell that I'm going to focus on set theory, I didn't know that when I first started my journey in math. How did I learn that? By tasting each topic and choosing my favourite flavour.

4. The previous point brings me to this one: my dad was in the academy for many years (though in history) and he told me before I started my degree that the first degree is horizontal. You learn a little bit about most things, on the second degree you start focusing on some topic, and in your Ph.D. you study the tiny iota of something. But you need a wide base for that.

5. One lecture in number theory we had some other professor to fill in for the regular teacher of the course. He was talking about how there are infinitely many primes of some form, and he said that there is an iff theorem about it, but we'll only prove the simpler direction. At one point during the proof one of the students (the majority of the class was computer science majors, not math majors like myself) raised his hand and asked if we can't use the other direction of the theorem. The professor answered that "This is mathematics. You don't use things that you haven't proved." and in a way he's right. Especially when you're only taking your first steps. It's important not to skip too far.

6. Last (but not least) I'd like to partially repeat several of the other points that I made. When I was a freshman, I came up with some idea and shown it to my linear algebra teacher. He was very impressed because I came up with it completely by myself (that was $||\cdot ||_\sup$ norm on $\mathbb{R}^n$ and definitions for metric, etc). He directed me to several topics that I might read about and learn more: topology, functional analysis, and a few more. But he strongly repeated that you make stepping stones in your way to the knowledge, and when you make them too far apart you'll eventually fall down. And he was right - I did fall down several times because I did that.

So we're being taught all that basic mathematics because we ought to know at least some of it, meddling with it helps to develop a sense of intuition about math and of course the abstract thinking process. Moreover, it's good to let the children play with theorems that were ground to dust and cleaned out of possibly mistakes, rather than new cutting edge concepts that might have problems and unclean environments that need extra-care.

And of course, even now when I finish my first degree I can't remember over half of the things I studied a lot for. But I remember the intuition and I have the tools to rediscover the knowledge when I need it.

Hopefully - never :)