How could I possibly have seen that it was a cultural heritage issue when you explicitly denied that that was the case? I'm trying to explain how I see the situation, but don't mistake that for me being an ideologue; I am legitimately interested in being refuted, here. Because a lot of people who seem to have a pretty decent grasp of how to behave care very deeply about this, which suggests to me that there's probably some worthwhile reason here. I just haven't heard it put to words.
So, again, what's the argument here? Is it the truth of language? Is it protecting a cultural heritage? Is it economic gain? Pick one or all or some other argument and support it. Write an essay, not a speech; I don't want any of this "how would you explain" stuff.
I'd explain the same way that I would that they have no monopoly on the words "cheese", "wine", or "beer" or any language's variants of them; I'd explain that in places other than their own, their location's goods were in the past so popular that they became incorporated into the very language, so that to the American speaker "champagne" literally means "sparkling wine". I would congratulate their ancestors on having such a profound impact on our own culture, and thank them for their continued dedication to the craft, and for their superior quality to our knockoffs, but I wouldn't destroy that contribution for the sake of salving their egos today. To do so would be an insult to the ancestors they claim to venerate!
But we can spend all day on platitudes like this. Make an argument, not an appeal to an emotional attachment I obviously don't have, for if I did I wouldn't be asking about this.
Convince me. The world does not turn on Europe's opinions, any more than it does America's, nor should either of them have the power to create a global monoculture.
The problem with enforcing the geographical indication only for countries where they hold any connotation is... well, first, how the hell do you enforce whether a term carries connotations or not in a country, and two, this would require the changing of names of a product depending on where it is sold. That is logistical hell.
It'd be fairly reasonable to commission a survey for any product that a group cares to press suit on, and changing the name of a product depending on where it is sold can't be that much of a logistical nightmare because businesses already do that. It's called translation. It's often required for labels and such so that the information on them can be read by local people. In fact, many products have entirely different packaging, or even different recipes!