Well, if we strictly follow Hyatt T&C, the above still follows the logic.

We use the CC to pay for the room. After the guest checks in, the room is considered paid. The actual guest can present the same or different CC to pay for hotel incidental charges.

At check-out, the guest receives the BILL and all the charges. Room has been paid for. Accidental charges are paid by the guest. It is the actual guest who receives stay credit and hotel points.