The way it's stated is by far the most likely story. Finished obsidian goods were typically produced relatively close to the mines because stone is heavy and even the best artisans frequently screw up obsidian pieces. Working with it also takes years to master. It's unlikely that a European artisan would have been replicating the style that early.
At some point we have to invoke the principle of parsimony. Europe did not have a tradition of obsidian mirrors in the late middle ages/early modern, much less ones carved in mesoamerican styles.
Yes, it's not impossible that it could have been produced in Europe, but it's far more likely that it was produced like the numerous other similar mirrors dated before contact and documented as produced by American artisans in early writings. This is all the more likely given the known supply chain for such objects, specialists in Aztec mirror production are specifically known to have used material from the two sources identified, and that Spaniards documented shipping such mirrors back to Europe as gifts.
There was actually a moderate amount of worked obsidian produced in the British isles during the neolithic. The only major source I'm aware of for it was Arran. So-called Arran pitchstone is pretty ugly though. Not relevant to the objections, just a fun fact.
