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?