Visa and co. mostly don’t work on modern mobile phones.
Never had a problem. Not sure what that means
It means that you can't transmit card details to a terminal through NFC. In general, Apple, Google, Xiaomi and co. push their own payment solutions (Apple pay, Xiaomi pay) and block use of NFC by other parties.
The networks don’t do credit, or cash back rewards, points and the like. That’s not Visa and friends. Those are offered by the banks who back the accounts.
I was using a short-hand language with the assumption that You understand that we are not talking about technological implementation. So to unpack my statements for You: Visa and co. mandate that banks and other financial providers implement credit, cash back or customer protection. ECB in their proposal of D€ does not mandate such functionality. In fact, the current proposal prevents implementation of such features. In sum, D€ is a different financial scheme than credit card schemes by Visa and co. and D€ is the less costly (not mandatory implementation of credit, cash back etc.) and more privacy-oriented scheme of the two.
Debit cards don’t have those options and work exactly the same as far as the charge network is concerned.
You listed AmEx previously and AmEx doesn't do debit cards. So I assumed You want to talk about credit card networks. Visa and Mastercard's "debit" cards are deferred-debit cards. Specification of D€ is more akin to prepaid cards. So again, there are differences and I'm not sure what kind of "public network" You are arguing for.
In general, Visa and co. provide in parallel a large set of (often competing) financial transaction tools. Mostly, they haven't invented anything, they just try to stay relevant in the rapidly evolving environment of financial-transaction providers. Especially, in case of digital wallets they are lagging behind. ECB can't afford that.