Privacy.com issues cards from the Visa and Mastercard networks.
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.