Proofs are only an example of what grade school teachers don't know. They are obviously missing all of the modern applications as well. Whether you want to call those applications "math" or something more specific ("engineering", "cryptography", etc) is a matter of taste I guess. The point still stands that kids are learning hardly anything of theory OR practice in their grade schools, and this is due to the teachers not knowing those things either.
We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.
The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with MULTICOLORED (whoa!) text. There is no communication between modern mathematicians and grade school teachers in America; they may as well live on separate planets. Math classes in American grade school are little more than busy work meant to weed out people who hate busy work. Infer from that what you will.
We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.