I'm sure this has happened to you: you find a charming restaurant, get all excited about the menu, the ambiance, the "there's something about them" staff and decide to dine at the establishment. Then you're hit with the cash-only edict that you missed earlier. If you really want to eat at that restaurant, an ATM run is usually the next step. Do you ask that they accept your payment via credit card? I have never witnessed someone try to convince a cash-only restaurant to accept their credit or debit card. Perhaps it's happened but I've never been privy to that. Once you've agreed to enter into a business relationship with that restaurant you pay according to their terms.
If your boss told you, "I'm not comfortable signing your checks so I will pay you with a gift card", would that go over well with you?
Didn't think so.