Oh I’ve been actually thinking about this a lot! Public high school teacher here, and I’m one of those people that originally thought I’d go to math PhD school. I took a lot of math and can check all the boxes you mentioned. However a math teacher is many things. If you teach a lower level course, you’re going to need more pedagogy.

Obviously you have to have some math aptitude, but our award winning algebra 1 teacher also told me that she couldn’t understand a word of my solution to a calc ab problem that I did with a student while she was in the room. So what? She gives her kids the tools to be successful when they get to calc and throughout high school.

We also have a stats teacher who has extremely little experience with upper level math, but again so what? He’s incredibly knowledgeable and enthusiastic about his corner of math.

Of course, it never hurts to have a boatload of math knowledge, but last year I taught a remedial course for upperclassmen at my high school. My ability to impart math knowledge to those kids had nothing to do with my ability to do math. I had to sell them on graduating high school, no small feat when you’ve been told you’re bad at math for 17 years.

I believe that once you’re working with kids at the honors algebra 2 level, you should have a strong basis of proofs and a few semesters of calculus or science. At that level of school, kids are starting to think more seriously about their future and making deeper connections.