European banks definitely issue Visa and Mastercard cards (and in many countries, there isn't even any local alternative for card payments).
But given the network effect, Visa and Mastercard have a lot of power over both issuing and acquiring banks. Issuers have at least a choice out of these two; acquirers just have to accept whatever their customer puts on the counter at payment time (or lose ~50% of their card-paying customers).
In other words, participation is (relatively) easy; getting some amount of control, let alone autonomy, is almost impossible – again due to the network effect.
But given the network effect, Visa and Mastercard have a lot of power over both issuing and acquiring banks. Issuers have at least a choice out of these two; acquirers just have to accept whatever their customer puts on the counter at payment time (or lose ~50% of their card-paying customers).
In other words, participation is (relatively) easy; getting some amount of control, let alone autonomy, is almost impossible – again due to the network effect.