Given how common no-shows have become, I can see their point. Sometimes people make reservations at 3 or 4 restaurants and then decide on that day what they feel like eating.

I assume they are typing the number into their POS to see if it's valid, so they can charge you the no-show fee. I don't object to this in principle, but the whole thing seems a bit insecure to me. You can always give them a virtual card number, or buy a Visa gift card and use that, if you're concerned about your card number getting compromised.