I'm usually asked for my card overseas, not in US. I don't have a problem w/ them asking for the card, but do draw the line at them xeroxing the card. Yes I'm talking to you Prague Marriott.
FWIW - I want them to either store the card so I don't have to type it in all the time or ask for the card (or mobile app) upon check-in. Don't really care about either.
Cheers.