I believe Google Wallet works with any credit card because the payments go through Google, and then Google bills your card.
NFC tokenized payments (e.g., Apple Pay and Android Pay) use an alternative credit card number issued by the bank. Payments do not go through Google or Apple, and only the bank knows that this alternate number is associated with your credit card account.
Google Wallet actually moved to a prepaid model - you can only spend money that you add.
Android Pay works for me the way that you described Google Wallet working. I add a card number (seems like almost any card works, I have a MasterCard Amex and Visa Credit and Debit that all work) After I tap to pay Google charges my card for the amount.
EDIT: I just looked back at one of the recent tap-to-pay purchases I made with Android Pay, the charge on my card says 'GOOGNFC*[merchant-name]'. I believe that means that Android Pay charges are going through Google.
You have the old grandfathered cards (if you click through details, it will say it's a virtual card processed by Bancorp)
You can't add any new cards to Android Pay unless the bank supports tokenization, and payments are processed directly by them, no longer going through Google.
NFC tokenized payments (e.g., Apple Pay and Android Pay) use an alternative credit card number issued by the bank. Payments do not go through Google or Apple, and only the bank knows that this alternate number is associated with your credit card account.