Technology-wise this is a solved problem. Uses open-source software too. All the EU needs to do is say 'self-custody is fine, please use a EUR-based stable coin for payments EU-wide'. The market will do the rest (the stable-coin, payment terminals, integrations with cash registers, DeFi forex for places where EUR is not used, etc.).
I hope there will be more adoption of EU stable coins like https://monerium.com, and a payments ecosystem will form around it. The rails are there, we need the apps and the integrations next.