A closing protection letter is the underwriter saying to the lender, and in many states to the buyer and seller, that if their own approved agent mishandles the closing, the underwriter will cover certain losses. It covers things like the agent failing to follow written closing instructions, or fraud or dishonesty by the agent in handling funds and documents.
So you're right that it's about the settlement agent rather than the property. The title policies answer the question of who owns this and what's against it. The CPL answers the question of what happens if the person holding everyone's money at the closing table does something wrong with it.
That's why the lender wants one on nearly every financed transaction. They're wiring a large sum to a small independent agency they've never met, on the strength of that agency being appointed by a national underwriter. The CPL is the underwriter standing behind that appointment.
The separate fee exists because it's a separate instrument with its own charge, and in most states the agent remits that fee to the underwriter in full rather than keeping a share. Availability, wording, and who can be named vary by state, and some states set the CPL fee by regulation, so check how your state handles it before the exam and definitely before you're the one issuing them.
One thing that trips people up: the CPL doesn't cover the agent going out of business generally, or bad judgment about title. It's aimed at the closing conduct.