It depends on your payment gateway. I can setup a POS payment system with any payment gateway processor I want. Depending on the system you can essentially run direct deposit into Google Pay, and as a vendor I can direct deposit the money into my checking via my gateway.
But that's not the issue being contested because there are hundreds of payment gateway processors available.
Right, you CAN set that up, but in the US, not accepting visa and Mastercard is essentially untenable if you are a retailer or restaurant. You might be able to avoid accepting AmEx, but from talking with restaurant owners, the biggest spenders are often AmEx holders and are annoyed at having to use a backup card.
But that's not the issue being contested because there are hundreds of payment gateway processors available.