Is this really the end of the credit card? It sounds like an abstraction layer for payments. The backing money used to pay will still be some sort of payment source like a credit card or debitable bank account.
As you said, "credit card or debitable bank account"; you already mentioned one way this could avoid a credit card. Now consider that Square controls both ends of the transaction. So, any payment source Square chooses to support will work: credit/debit card, bank account, some pre-paid account, line of credit with Square for which you get a bill later...