It would be nice if I'm going to give my card details to over the Hyatt app or their website that, they charge it and keep it on file for additional incidentals right when I make the reservation. I don't pick rooms which require a deposit and would rather just pay in full at the time of making the reservation right then and now. If they are worried about somebody making unauthorized charges on my card or to ensure that the guest is the person who made the reservation, then that's why I show ID.

I normally don't use my cards when I visit Mexico and just pay in cash, but it seems like in my experience Hyatt has asked me for the physical card in both the US and Mexico. In Mexico, I just pay in cash for my stays or use Orbitz prepay for my stays.