I think that the people who teach the kids to play football there are called "football coaches" not "PE teachers".
In BrE, a PE teacher is someone who teaches children from the age of 5 until about 12, teaching them a variety of sports or just helping them to keep fit. My PE teachers in my first two schools took us for swimming, netball, hockey, football, kickball, gym etc. They did not usually have a specialist sport.
In America, I believe that sports are taken much more seriously in schools so there will be a specific baseball coach, football coach, basketball coach etc.
As always, I'm ready to be corrected by my friends across the pond.