As the saying goes, “A rose by any other name would smell as sweet”

EU law and its constitution have primacy.

Sort of. It's not quite like anything else. The EU has mostly trade and standardization powers, and lots of the other stuff is an outgrowth of that.

But mostly, EU law just sets a baseline, and almost all execution of it is devolved to the member states.

Edit: the EU does not have a constitution but a constitution shaped update of the Treaties. Now, lots of politicians are happy to blame the EU for unpopular stuff, but the council is the national politicians and they basically run the EU.

I'm not going down this semantic rabbit hole, which is way beside the point. :)

It's not a semantic rabbit hole, it's critically important if you want to understand how the EU works, and who actually has power.

The EU is not a state. Personally, I'd prefer a much more federal EU, but that won't happen anytime soon.