Bottom line up front: how do I deal with directs' perception that my boss is clueless in our field and gives bad direction - when I agree with them - without seeming grumbly and disrespectful/unsupportive towards boss?
More: Newish role, newish manager. I am the senior person in the company in my field of expertise, and at interview stage, both my boss and the CEO were upfront that my boss - also new to the company - had zero background in my field and would look to me to run that area.
My boss is clearly outstandingly experienced and extremely competent in an area where I am completely clueless, and I respect and admire his abilities. However he frequently gives direction on what I should do in certain situations, and to be blunt, his direction and advice is bad. It's like putting an orthodontist in charge of a hospital then having him tell the emergency physician how to save the patient that came in in an ambulance.
Perhaps the one thing that has saved us from disaster (or dead bodies and malpractice suits!) is that as an "emergency physician" my work requires rapid decisions and actions, and usually by the time the "orthodontist" weighs in, the patient is already out of surgery.
This is not great, and I wonder how sustainable it is, but I could live with it.
Where it gets tricky is that he frequently gives his advice/direction in front of my directs. I am blessed with terrific directs, all with 12-25 years experience in various areas of "emergency medicine", and I have worked very hard at these relationships. It's clear to me that they see what I see. It has a direct impact on them, and I am not sure how to handle it with them. On the one hand, I don't want to criticise the boss to my directs; on the other hand I want to help them to understand how best to work with him and I want to help them get results, and that's difficult to do without allowing them to discuss the challenges they face in their dealings with him in an open way with me.

It wasn't completely clear
It wasn't completely clear to me given the quotation marks, is the medical/surgery part an analogy or reality? Because if a life is at risk due to bad direction from your superior that trumps any other obligations you have, and you should act weeks ago to change that.
re: managing directs comments on my boss
I understood the medical/surgery as an analogy. Understanding it as an analogy I don´t think I would worry much about it. It seems like you boss is smart and knows other staff you don´t know. Your boss doesn´t need to know all the staff you do. And that is the message you must pass on to your direct, your admiration and respect to him in all areas he is good at. And reinforce the idea that none of us know it all. And that he is sharing with you and that you should also share with him.
With respect to challenging the directions he gives I guess the best think you do is work on your persuasion skills and also understand yours and his DISC profile. I your boss perhaps and I, or a D? I´s think talking, not everything they say, or not every direction they give is the definite direction. D´s may arrive to a conclusion too quickly. But they may have no problem admitting they were wrong and moving on into a different direction. I think it is wise that you do not challenge your boss´s direction in front of your directs. But you can talk to him afterwords in private and perhaps persuade him to work at it differently. And then you inform your directs that you have discuss the issue further with your boss and after taking more aspects in consideration the following has been the decision made. If the direction is life-or-dead and urgent, of course, you must challenge your boss´s direction. There are ways to do it, like: "that is one option. We can also do it this way which have this positive impacts.... Would you agree?" If your boss is a high D-I, he may as well agree.
Are you a high C? Specially if you are, you must remember: "The other way works just fine too." Are you open to the idea that perhaps your boss´s suggestion are not as stupid as you think? What specific disaster are your really talking about. Are you sure it is a disaster? Or is it your view of it? And also taking the DISC profiles in perspective, are you sure it is a direction of where to go, or your boss is sort of brainstorming with you (seems like you haven´t been following his direction, or else everything will be disaster by now!), and there is room to discuss back the orientation he is giving (up to a certain point).
In short, I would neglect any impact of the team thinking the boss is stupid, he is not, and this is the message you must pass forward, and I would focus on what specifically to do with the directions he gives and for that you must understand DISC, yours and his to evaluate how you argue and how you team up with your boss. Once you understand how you team up with your boss you can explain to your directs what they should do (should they argue with him, do they give him facts, or do they just follow his orders?).
Nara