Like others, I have also been asked for the physical card - either at contract lounges, or in countries where the culture is that physical tokens and paperwork are preferred over electronic records.
On-your-device-only cards may work fine in the USA, but AA's customers in the more distant (geography and culture) parts of the world will have problems.
Now, making the card on-request might be a good compromise - people who travel only in the US might not need it, but those of us who get about more widely can request one.