They're protected from being cut out because they are the networks on which the credit card system runs. If you look at a stripe-issued card, you'll see there's still a visa or mc logo in the corner.
see e.g. https://stripe.com/us/payments and https://stripe.com/connect (under "Rigorous Compliance" and "Compliance respectively").
You'll also find them on CA's directory of money transmitters. http://www.dbo.ca.gov/Licensees/money_transmitters/money_tra...
That currently happens when an issuer card is used in a merchant that is acquired/processed with the same issuing bank. It is called an "on us" transaction and doesn't need to fire through the card scheme rails.