What you're missing is the erosion of the ability of the executing states to say things like "hey this is sketchy, we think this crime might not have happened", "hey the police department in this particular city is notoriously untrustworthy", or "hey this prosecutor is widely known in the local press to be corrupt and owns a collection of ferraris".

Now foreign authorities are trusted by default and significant parts of their reasoning are not subject to review, that's bad.