Those free bags, aren't really free.
Each time that benefit is triggered AS charges BofA for the bag. Obviously it's not as much as a guest with no status/card pays but it's a good chunk of change for the airline when all is added up. Hence it needs to tie back to the card number. At least that's how things worked a few years back and I don't see why that would have changed.
Airlines tend to know which of their members have one of their credit cards. I don’t know for sure about AS (other than that they *should* know), but for example AA, UA, and DL offer various benefits to cardholders that don’t involve *using* the card for a transaction.
I don’t know if AS charges BofA for a bag if the ticket was charged with the BofA card - what would they do with elite members who get a free bag? I think this is just a way to drive charges to the BofA card, just like the requirement to use the BofA card for companion cert tickets or the $6k spend requirement to earn such a cert. Airlines don’t like it that we use cards with better earnings or benefits (CSR or Amex Plat).