Both profit centers and cost centers are supposed to increase profit, otherwise you wouldn't hire those people. If a department doesn't drive profit, then just axe it and stop wasting money on it. If you can't [1], then it actually drives profit, you just have no idea how. The fact that you can't set proper KPIs to measure impact doesn't imply absence of impact, and of course it doesn't imply that you should treat it as if it has no impact.
Departments are engaged either in primary or support activities. And those activities are either efficient and optimal or not. Your competitive advantage doesn't even have to be your primary activity. If you are successful at making fidgets because you hire, plan and budget better than others in your industry; what insight can talking about profit centers and cost centers provide to you?
[1] How is someone supposed to run a company without HR or accounting?
Everything the company does should generate value.
Of course it does.
Whatever your business leaders believe is reality in that company. It makes no difference what you or the outside world believe.
You're either OK with it, or you need to get out ASAP.
In other words the business as a whole should be viewed as a singular profit center, and each department or project should be evaluated in terms of how much it contributes to optimizing the overall performance.
Well, those are rather funny categories in the first place. Cost and profit aren't complementary categories; cost and revenue are. Profit consists of them both (i.e. the difference between their sums across all departments).
Just as there are no pure revenue centers -- every department has costs! -- it could be argued that there are no pure cost centers: Every department contributes to revenue in its own way. Without maintenance, you'd soon have no services to sell; without R&D you'd soon have no new products to sell... Heck, even HR -- without it, you'd soon have no personnel to staff the other departments.
Seems a bit skewed that some departments get to hog the "profit" label all to themselves while others are -- pejoratively, in MBA-speak -- condemned to the "cost" category.
It doesn't matter how many presentations and case studies and financial models you build, whether you're a cost center or not is basically entirely determined by what a few people in the c-suite believe. This became blatantly clear over the past year or so of layoffs, when the same people who nodded and thanked teams for showing them how they generate revenue for the business immediately turned around and put them on the chopping block.