I have a credit card from Wells Fargo. It's still made by Visa and Visa takes its cut of credit card processing fees.
My point is that in practice you can't issue a credit card and avoid Visa (or Mastercard) processing fees.
You would have to get most merchants to upgrade their POS terminals to support your payment network and that's not going to happen.
Technically there are 26 different cards most POS terminals support. 11 of those are Visa or MasterCard
Edit: "Private labeled" is considered anything a normal gateway will accept and go into the generic category. Which absolutely exists
I have a credit card from Wells Fargo. It's still made by Visa and Visa takes its cut of credit card processing fees.
My point is that in practice you can't issue a credit card and avoid Visa (or Mastercard) processing fees.
You would have to get most merchants to upgrade their POS terminals to support your payment network and that's not going to happen.