Why do you need to study a subject at a university level to teach it at a high school?
On one hand, when I myself was in high school I was wondering about the same thing - especially that I learnt the whole high school material (in maths and physics, I mean) in the first year, so felt that I could have taught as well as a teacher. But indeed, as pointed out by Xander Henderson and Massimo Ortolano, only at the university (in fact, only during my PhD studies; well, in fact only as a post doc; well, in fact... hm, you get the idea) I learnt the bigger picture.
So, studies are so that if a student asks "what will I need the quadratic equation for?" you could give a better answer than "because it will be on the test".
What is a better answer in this case is a whole different issue. I have lots of friends who indeed, once they finished high school, never solved a quadratic equation again. But on the other hand, they also didn't need the knowledge about Hamlet, genetics, WW II history, etc., in everyday life. That doesn't mean that they, we, you shouldn't learn about those things. I have lots of knowledge that I don't need in my life - but I enjoy obtaining it, possessing and connecting it with other areas.
History and experience shows that those who do not have advanced training in, e.g., mathematics, and try to teach or write about it, give presentations that are somewhere between hilariously and atrociously wrong, broken, mangled, incorrect, damaging, and misleading. Secondary institutions can partially mask this by teaching to standardized tests (esp., multiple-choice ones), but the damage to actual understanding is still done, and people suffer from it later.
For an example attempt at remediating some of these effects, consider Hung-Hsi Wu's papers on "Teaching School Mathematics".
In short: If somebody doesn't like the idea of studying their subject at a university for a few years, I really don't like the idea of that person teaching children (in addition to the pragmatic reasons given by others).
Personally, what I remember most about math in high school was that my teachers were enthusiastic about the what they did (despite going into math, I don't recall learning any particular thing in high school; just that I got the standard fare).
I certainly would prefer students learn from somebody who at least enjoys their subject. While "study subject X for Y additional years" certainly doesn't guarantee that the person likes subject X, I'd hope that it discourages those who don't like the subject from pursuing this path.
I say this because I find few thoughts more depressing than thinking about the experience of a child learning from somebody who fears or resents the subject. And yet, it happens -- I've seen education majors that don't like or don't understand basic mathematics which they may some day teach...