I've heard that too (though not as part of any official education), but I'm not sure I really like it. As far as I know, men have always had the upper hand in the part of history that shaped most of our current society. When prompted about it, it's easy to go "oh no, I didn't mean it has to be a man, I meant it generally", while the word still obviously refers to a male. It almost certainly reinforces an expectation of male leadership. I sometimes use the 'wrong' gender on purpose ("she" when talking about male-dominated professions and vice versa), just to make the reader realize that's an option, hopefully getting used to the idea and not frowning when they hear of an actual case where an uncommonly gendered person is (studying for / a candidate for) $profession/$role.

The high school idiom is not inclusive. I agree that 3PP is rarely felicitous to those who speak English. In this case it contributes to confusion: "they" refers most naturally to the entire organization. The best alternative depends on the context. In this case one could have written, "That person is the boss."