COVID-19 has proven that if anything, the European Union tends to spread national initiatives among other countries (and Germany is often a leader in EU).
In this specific case, the EU is more likely to be the type of organization that would think about how to create a unified permit
-> as they did with the EU Digital COVID certificate; some sort of "I am in the register of mobilization" / "have a temporary travel authorization".
So, EU might be an enemy that pretends to be your friend there.