They connect to Visa’s server which authenticates the transaction. They’re the network in the middle of the system that everyone else connects to. The banks just provide the account to transact from.

That’s incorrect. Using push-up messages, the authentication is done in bank’s authentication app. If using SMS authentication, the SMS with code is sent and signed by bank. Visa and co. provide a bit of infrastructure and the messages between the processor and customer’s bank travel through their network, but this step is not essential for the transaction. For instance, processor could directly communicate with customer’s bank through ECB’s open banking API.

Then how is this new Digital Euro supposed to do NFC transactions if they can’t use the NFC?

Through regulatory effort. As part of WERO initiative it is already being discussed how EU will force Apple and Google to open their NFC chip. Visa and co. may piggy-back on that effort, but crucially, the regulation will be designed to facilitate implementation and adoption of WERO and of D€ and not the use and adoption of debit cards.