But to keep the analogy going, the decision of whether to keep doing business with them is not with the person who relies on them to supply appropriate goods in a defined timeframe, unless you want to make an org chart where someone has multiple bosses.
You can solve issues of teams being ineffective, poorly managed, or issues with individual performance without trying to force every working relationship within a company into a client/provider agreement and build up a ridiculously complex internal market with none of the freedoms that usually exist on both sides of the client/provider relationship.
Difficulties in getting rid of underperforming staff usually result from companies taking a very lax attitude to setting achievable objectives and then evaluating staff against those targets - usually because it's a double-edged sword where it becomes difficult to explain why you aren't going to award a payrise or a bonus to someone exceeding all targets. If you actually bother to tell an employee what is expected of them, and those requirements are reasonable, then if you follow the proper processes ideally that person ends up improving. If they don't then you keep following your process until eventually they can be dismissed. If a company wants to take short-cuts and just get rid of someone then sorry, but it's absolutely right that there can be redress in that instance.