The EU is definitely the entity responsible for acquiring the vaccines on behalf of nation states - this is what is failing.
The 'delay' is not due to Denmark inability to field nurses, it's in the EU's ability to secure product - for whatever reason.
And in the end, it won't matter. If the EU ends up 6 months behind other nations 'there will be no excuses'.
I think that they'll probably catch up, but there's a real possibility they don't.
Why? This isn't something the EU normally does.
The approach actually makes sense, if everyone was acting rationally and competently it would work.
But they screwed it up.
Everyone is clamouring about the 'new world order' and 'pivot to Asia' and wondering what the 'Brexit' result would be ... and the optics of this are 'EU in last place'. It's just a bad look, and it's an event that will ring through time with long lasting effects, at least in 'hearts and minds'. There's still chance to catch up however.
For example Denmark is doing quite well, but Bulgaria is doing quite bad.
So to repeat, healthcare is not a responsibility of the EU.