I... have wanted to do this. I have not, in part because I can't afford someone with middle-management experience, and in part because I don't have the middle-management experience required to tell if someone is good at middle-management.
We've had many discussions about this within the company. My buddy suggested a 'babysitter' while my dad suggested a 'mom'
But yeah, that's the thing... middle-management can be effective as 'chair kickers' - Hey, Luke, get back to work. but... I think that middle management generally gets in the way when it comes to decision making... and there's no way I'm letting a non-technical hire (or fire) technical people.
Now, other people have said that what you are proposing is common in the business world; they are called secretaries. But... I think that's a different dynamic. As the other comment said, "more like an olympic coach" - I don't want someone to make me more comfortable... I want someone to push me. I mean, especially when I hire people who aren't good friends already? there's a problem. They don't want to say bad things to me about my behavior. They'll criticize other employees, but not me. I mean, I'm sure it's a problem that can be overcome.
Of course, I know other people who just don't need external guidance. There really isn't any need for middle management with those folks... they just do it. For those people? Management is largely unnecessary. Unfortunately, I seem to be in the majority here; I get a lot more done under a manager, usually; though personality-wise, like many technical people, I chafe but hard when someone I see as less technically capable than myself tells me what to do. It's a problem I need to solve... or I need to start working for other people again. You can only coast on the strengths of your hires for so long (especially if you can't pay enough to /keep/ those hires for that long.)
Actually, thinking about this, I think I might actually institute a 'person X is middle manager for day Y on department Z' policy with people I have now, where X isn't a particularly senior person.