While I can understand that a new manifest would be needed (and as I understand it CBP charges for processing the manifests, so that expense would absolutely be passed on to the passenger involved) I don't believe that it changes the closed loop nature of the cruise for the remainder of the passengers. There are undoubtedly some passengers onboard traveling with government issued ID and birth certificate and if the cruise was actually deemed to be non-closed loop then they would need to be disembarked at the next port, since every passenger needs to maintain the required travel documentation throughout the cruise and it is illegal for the cruise line to transport them without it. They also wouldn't have the required documents for any downstream port visits.
In any event, even if the authorities did somehow view the cruise as non-closed loop from that point on since those with birth cert/ID are not disembarked then some exception needs to be made to allow them to remain so from a practical viewpoint it would hardly matter how anyone views the cruise. (The same scenario would arise if someone missed the ship or left the ship due to emergency, so this isn't something that only happens when a passenger decides to leave the ship at a port as suggested by the OP.)