Not sure what to make of the claim that Hungary might theoretically be enforcing Hungarian law in France. It seems surprising that no-one has noticed any specific consequences of this that you can point to.
The EIO is mostly just a formalization and standardization of a bunch of ad-hoc processes that were already in place. Law enforcement agencies in different European countries do try to assist each other, on the whole.
Now foreign authorities are trusted by default and significant parts of their reasoning are not subject to review, that's bad.