A reported collection is the reporting of the fact that, at some point in time, they had collection authority on the debt.
Satisfaction of the debt is not basis for deletion of the collection, regardless of whose hands the $ passes through.
If the debt is paid, the collection is terminated, and thus the debt collector must update the balance under collection to $0 and close the collection.
The only deletion "requirment" is not under the FCRA, but only an admin reporting procedure of the CRAs that they apply only to cases where the debt collector has had their collection authority terminated prior to payment of the debt. In those cases, the CRAs have established a reporting policy that the debt collector delete their collection, with the intent of the policy to then prevent a second, and thus plural, collection from simultaneously reportting on the same debt.
In cases where the debt is paid while the debt collector still has active collection authority, CRA policy is clearly stated that the debt collector not delete their reporting.
If the debt remains unpaid,, then deletion of the collection can be pursued by offering a PFD to the debt collector.
Acceptance of a PFD is voluntary on their part, and is contrary to the reporing procedures set forth by the CRAs.
