It would be like if a US company registered Champagne as a trademark then enforced it against French companies selling it.
Does non-PDO Parmesan exist in Italy? In the US, "Parmesan" is usually some kind of domestic hard cheese that lacks maturity. The genuine Parmigiano Reggiano costs around 3x.
While some California wines were and still are sold as "California Champagne", there was a court decision back in 2006 which affirmed the same protected trademark for Champagne as in the rest of the world, with the caveat that wineries already using the "Champagne" name would be grandfathered in and permitted to continue doing so.
A bottle of Champagne made in the US will be labeled with its origin, and no one can start a winery and start selling Champagne in the US. That's not the same thing as "generic".
https://www.ttb.gov/system/files/images/pdfs/rulings/2008-1....
But yeah calling anything Champagne apart from stuff coming from Champagne is a bad form and looked down upon a bit. Small kids do such mistakes and thats fine, adults shouldn't.
You will get an entirely different style of wine.
The origin location is really secondary. “Franciacorta” gets you something like Champagne, “sparkling wine” does not.
And no, "trademark" definition aside, it's not "generic" in the US by any definition of "generic" commonly used in the US, legal or otherwise.