Best case, the physical card, if they have one at all, will be a pre-paid gift card, not a card linked to your account. And the virtual card ... better than a physical card! ... will be a virtual gift card.
Worst case, they've had so much trouble getting companies to join the trainwreck that they're going to try to claim that the Chrome extension is, in effect, a virtual credit card in some sense that only the hardcore GAWbots will try to believe. And better than a physical card!
To be fair, an actual virtual card - especially if it's a disposable one-time Visa or MC card number that you generate for a specific purchase - would be an interesting solution. Almost no risk of fraud, fast, and solves the acceptance problem. But I'm quite sure they will somehow manage to screw it up with their KYC requirements, "floor", Premier, and general ineptitude.
if this was so easy/desirable/feasible then Gyft would do it because they are owned by Firstdata who is in bed with the credit card associations.