I've had issues with this on 2 DL flights on which I needed to check a bag over the past 4-5 months.

When flying DL I typically have my Alaska MVP Gold (not entitled to free checked bags on DL) number on the reservation but both times I was told at check-in that they were unable to waive the fee despite my having my physical Alitalia elite card on hand. Only after putting my MilleMiglia number into the PNR were they able to waive the checked bag fee.

Based on this as well as similar things I've heard from others, my assumption has always been that getting the perks require having the actual elite FFN on the PNR. Just trying to clear this up as if this is not an actual policy, I'll be more insistent if I'm told that just having the physical card does not suffice.