Not European, but here's a structure that I think would be best (for all currencies, European or not).
Central bank creates a nationalised corporation "National payments processor". Loans out money to NPP to create a copyleft MasterCard competitor.
NPP's objectives are to reduce interchange fees, establish sovereignty in this space while keeping transactions secure.
Now, from what I understand, a retailer cannot charge different rates for different payment processors. Meaning, if I am a retailer, I can't charge more to customers who pay using Amex (who have high interchange fees) compared to those who pay using visa/MasterCard (lower fees).
Meaning, if NPP keeps interchange fees low, the benefit would be passed on to retailers directly. Consumers would see 0 benefit. If consumers see 0 benefit, no one's going to pay using NPP. This is the case with interac in Canada. Interac payments are better for retailers. But I see 0 cashbacks through my interac card. Why should I not use my visa credit card instead that gives me better cashbacks?
Therefore, here's what NPP does: it charges marginally less interchange fees compared to visa Mastercard, WHILE passing most of the fees charged to the retailer directly to the consumer as direct cashback.
Consumer adoption happens because of better cashbacks, retailer adoption happens because there are people willing to pay using an NPP card (and also the sliiiightly less interchange fees).
Now, to the organisational structure of NPP. State owned corps are prone to corruption. Accountability structures are top down. If I, the taxpayer owner of NPP am seeing corruption in NPP, I have to threaten my MP with my vote, who has to threaten the PM with their vote, who then has to threaten the finance minister with their job, who then has to threaten the head of the central bank with their job, who then has to threaten the ceo of NPP with their job.
Instead, while the state maintains equity over NPP, the operations of NPP are controlled by a state started consumer cooperative, where member owners are those who own an NPP card. This way, accountability structures are much more direct. The inefficiencies of state owned corps are severely reduced while maintaining the benefits.