Visa and Mastercard are called card _networks_ for a reason. Wherever you are in the world, or in any site anywhere, if your card says Visa and the merchant’s POS machine (or payment gateway) take Visa, both parties know the transaction is good. The merchant gets his money and you get the product.

You get your card from your issuing bank, so the consumer’s last mile is the bank’s problem. The merchant get their POS/gateway from the acquirers. Your bank and the merchants acquirer don’t know each other.

Visa and Mastercard are intermediaries. There’s no way a NatWest card in the UK is connected to whatever POS is in Chile or whatever. They all route through the card brands.

This is why it’s so tough to break this monopoly.