There is scenes where it's shown that Cullen and Leliana are close to their subornates. I'm not saying that they are friends with Cullen and Leliana, but they know these persons. They don't even know Inquisitor that much and aren't in touch with Inquisitor either. The one commanding them isn't Inquisitor, it's Cullen and Leliana.
EDIT: I think it's like Inquisitor is country's president and Cullen it's army's commander and military is usually more eager to follow it's commander than president that is closer to them than distant figure head.
As i said this is completly impossible ,leliana and cullen may be close with few people she constantly deals with and keeps close to her ,as inquisitor may be close with advisors and companions but can't be close with people under his command ,same for advisers.As i said leliana may know about existance her spies and spies may know lelianas existance ,but majority of them at best just saw her and that is all.With cullen it is even worse because scope is even bigger.
As i said no in first place people are loyal to inquisitor and see him as religous figure ,in fact inquisitor is pretty much what unites inquisition in first place. If that was true entire armies would fell apart because change of military leaders ,what in fact can be true but only if such leader is seen as inquisitor and is only thing that unites that army ,cullen isn't such person.