I understand that we're mostly agreeing, and I think that we both have an intuition for what it means for something to be a profit center.
Still, I'd like to show a gap in what you said: You talk about what kind of increase in revenue you'd have if such and such people did a better job, and what kind of ROI you'd get from each department.
But the question is, an increase relative to what, or an ROI relative to what?
Let's take the billing department. If you made sales for $10 million, and your billing department works okay, then you would have a revenue of $10 million. If you hired the best billing experts on earth, you still would get just $10 million. So that might make you look at the billing department as a cost center rather than a profit center.
But maybe if the billing department fucked up royally, either by not charging a customer, or overcharging them and hurting the relationship, or breaking obscure laws and getting the company in legal trouble. Then you could be out a lot more than $10 million.
So in effect, you could view the billing department's ROI as "how big a disaster they can prevent, and how good they are at preventing it." That kind of thinking makes sense for any department in the company.
That's the reason why I don't see the logic behind branding some departments as profit centers and some as cost centers.