The problem isn't just getting something that works across all European countries. It's getting something that works globally.
While we may make most of our payments within EU, basically everyone still occasionally pays for something outside of EU, either online or when they travel. This means if the new thing only works in EU, every European will still need and have a MasterCard/Visa even if they use it less often than before.
This is still a massive amount of leverage - MC/Visa still have the ability to block payments made from EU citizens/companies to outside.
You can buy things from your local Amazon or national equivalent that come from outside using this systems, so, you are not so restricted to EU sellers.
I suppose the most problematic would be traveling. I recently when outside the EU and was surprise how smooth the process was using my Visa card, to the point I didn't use any local currency.
On the other hand, I recently buy books from the UK and it get stuck for two weeks in customs, and it had nothing to do with the payment platform. I had not realized how difficult is to import something from outside the EU, even for personal use.
Also European merchants who need to accept payments from non-Europeans need to accept Visa and Mastercard.
Many (most? all?) of the payment systems I’ve used over the years can interop with Visa or Maestro. Case in point: my Bancontact cards can pay in any Belgian business even if they can’t afford the better machines that do VISA, but my card also has the VISA logo. Same in Portugal and Germany.