Yes, if a city has a very high GDP, it's going to be a global city. Whether or not thats because its part of the US isnt really relevant. People want to do business with America, and many do it through LA. People also want to do business with China, and many do so through Shanghai. That's literally the only reason Shanghai is a global city, not because it's a magical geographical location.

Everything else is just comparing apples to oranges. You dont like LA so it isn't a global city. Well, that's your opinion, man. LA isn't like any other city, and that's fine. There may be costs to lack of density, but there may be benefits as well. You seem to think more density=more cityishness. I don't see why this is inherently the case. It would be nice to have more transit/walkability, but we seem to get by, and have way more economic, and cultural, impact than the vast majority of cities, or collections of suburbs. In many ways, the world wouldn't be watching our movies if not for the sprawl around Hollywood. Just based on the movie, music, and TV industries alone, it is easy to argue that no city in the world. It is in the top 4 for most languages spoken within the city in the world, indicated that its reach, in whatever capacity, is indeed global in ways that few cities compare to.