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.
You can’t run your own card network easily because you would have to convince all of the merchant banks that take card transactions to do business with you.
Digital money movement requires an operating agreement between at least two financial entities - but most of the time there’s a lot more. Depending on the type of transaction you may have two or more gateways, facilitators, processors, issuers and underlying banks involved.
It’s a very fragmented system that relies on many, many different entities all having agreements and contracts with eachother.
Through Visa and Mastercard