If you've followed Dungy's career, you know that he has a very even-keep approach to coaching. He keeps calm and poised, and he wants his team to keep calm and poised. Instead of getting too up or too down about anything, Dungy keeps a professional approach, and it's that approach that I think accounts for Dungy's remarkable consistency in his coaching record. Stay calm, stay professional.
Profanity quite frequently (though not always) can be a break from that even-keel attitude. Some people and social groups swear quite casually (I've no problem with it). But think about what swear words often convey. Frustration. Anger. Perplexity. People often swear when they lose their temper, when they lose their cool, when they lose their calm poise. Furthermore, sometimes profanity isn't just an effect of losing cool, but can cause others to lose their cool. It's about atmosphere. It's about environment.
Dungy doesn't want to be around that, and if he's building an organization, he doesn't want the other members of the organization around that. He doesn't seem to say such people shouldn't coach; he's saying that if he is running an organization, he thinks that swearing creates a negative vibe. He says he wouldn't want his players around that. He's had a remarkably successful coaching career doing things his way; it's understandable that if he were running an organization, he'd want it run the way he's coached. It's not about the swearing: it's about maintaining positive, professional calm. And I think in this way I agree with Dungy: if I were in charge of running an organization, and I was hiring managers to plan strategies, work in pressure situations, and (most importantly) supervise, utilize, and motivate employees, I would find excessive profanity a turnoff.
I mean, think about this: when adults use profanity in their professional contexts, what does it usually mean? And how do other professionals respond to somebody using profanity? How do you respond to a superior swearing at you? The answers to these questions depend a great deal on your field, of course. And profanity is not inherently the problem (my friends, family, and even my students know I'm not a prude or an innocent when it comes to profanity). People can swear and still be even-keel, and people can refrain from swearing and be a complete up-and-down mess. But from a certain point of view, isn't there something a little silly about adult professionals swearing?* Is it possible it creates a certain unhinged atmosphere, where people are a little more on edge, a little more keen to give in to anger, frustration, or perplexity?
I think it is possible.
I also think critique of Dungy's stance (like Chris Chase's at Shutdown Corner) is valid. I think swearing can often be natural, can often be good-natured, and can often still come in an atmosphere and context of calm poise. I even think Rex Ryan knows what sort of environment he's trying to create with his team. But I can see where Tony Dungy is coming from.
* from another certain point of view, isn't it a bit silly to devote any angst whatsoever to adults swearing? Yes. Yes, it is.