If they used CC (not debit), then any payment would mean creating a debt, so yes, they would have to pay to the bank. Because bank already paid in their name.
That's why you don't pay large sums with CC, but with invoice + bank transfer, and have a limit set on your cards, when you do.
- control: you are in control, when you do the payment. You can plan your cash flow.
- additional advantages: You also have payment terms, some vendors offer discounts for earlier payments; if your cash flow can handle that, why would you giving up of that?
- liability: with CC, you are getting credit that is drawn at other party leisure. It's you, who is liable for this credit line, even if the other party made a mistake. You are always liable to the bank, never towards the vendors. With bank transfers, every single payment was authorized by you (where by 'you' I mean authorized person at your company) and the liability is towards the vendor, who is not likely to have such a strong position (see Porter's five forces).
- leverage: if another party makes a mistake, they have motivation to correct it. Every company in existence has already received invoices, that are incorrect. Withholding payment until they are corrected is a strong motivator. Without that, you could be left without invoices that can be put into accounting AND without money that you have to account for.
- setting up processes: when you grow beyond certain size, you are going to want to formalize both the procurement, accounts payable and treasury. Having purchasing and payment discipline that are compatible with that already in place will mean less pain from the growth, less things to change.
When we need people in the field purchasing small supplies, we don't want them to handle cash, so they get debit (not credit) cards, with relatively small limits. It is enough for them to get by, but not enough to make any damage of significance. (The exception is fuel and that's what fuel cards are for - basically it has a form factor of a credit or debit card, but works only for fuel, is paired to a license plate and the vendor sends invoice at the end of the month).
Another scenario, where CCs are useful, if you need to pay something right now; you don't or can't want to wait for the order->delivery+invoice->payment cycle. That's fine for consumer impulse purchases, but that should not be a normal way for company purchases.
Of course, if you start a new business relation, some companies would not trust you, that you are going to pay the invoice; sending advance invoice and paying it is fine. In practice, it is quite rare occurrence.
That's why you don't pay large sums with CC, but with invoice + bank transfer, and have a limit set on your cards, when you do.