You are drastically underestimating the value and control that the processor networks bring.
Wells Fargo might underwrite the credit on my credit card, but I would immediately cancel that line of credit if it wasn’t accessible through the Visa network.
The value of a credit card isn’t just the line of credit, it is the near universal and frictionless acceptance as a method of payment.
You're mostly right, but Discover specifically owns a bank. That bank does underwrite the debt on most Discover cards. Also Amex, but you didn't mention them.
Visa/MasterCard/Discover/etc are little more than clearing houses.
Banks serve as both issuers (to the consumer) and processors (to the merchant), often times both.