What it comes down to is that free checked baggage is a benefit of having the card. It is totally unrelated to how you pay for a given ticket, or who pays for it, or even if anyone pays for it at all (such as an award flight). That benefit goes with the cardholder when he or she is flying, plus travel companions per the card's terms and conditions, and is also unrelated to any other tickets that card may have been used to pay for.

Think of it that way, and the reason for all the above should be clearer.