For the record: I don't care whether women coach in the NFL. I sure as hell don't think women and men are "interchangeable." But your argument reminds me that men have always been allowed to coach women's sports. And somehow, Geno Auriemma and Bela Karolyi managed to scrape by without being able to "connect intuitively on a relational level" to life as a woman.
Putting aside football for a moment, imagine a girl growing up in another intense contact sport like lacrosse. Let's say her dad was a coach, and she was raised not only playing but immersed in sport with a detailed coaching mentality. She played from childhood all the way through college, then became a coaching assistant, then small-college head coach, then division 1 head coach. Are you ready to say that Amanda Belichick couldn't possibly succeed at coaching men's lacrosse and should be barred from trying?