I think Brexit highlighted this big hole. Who spoke for the EU? You'd occasionally see a headshot with a "EU President" title attached to it, which would then be dismissed in favour of the opinion of Merkel or Hollande who didn't even notionally represent the Brttish in any way.
If the EU is to govern so much, it needs a polity that Europeans jointly elect, with representatives that genuinely represent the EU, not local interests. The EU needs a face, parties with cross-border support, democracy, mandate... Even the UN has more of this than the EU. At least it has a face representing it.
I'm not sure this leads to better monetary policy (I don't really believe we confidently know what better monetary policy is), but I do think it's necessary if the EU is going to govern so much.