I think you may be judging Americans by British standards again. I know that we don't always tend to have names for supporters groups - although you only need to look at groups like 1894 to see that it does happen - but the very fact that every single MLS team's supporters groups have names and identities should suggest that for NYCFC there's no reason why it shouldn't happen either. It shouldn't really matter whether or not they name their SG though, since you as I are in the UK and will not be a member of it. Realistically, aside from members referring to it, you should barely hear the term used once the team gets playing; news of the games will drown out any references to it.
Otherwise, I somewhat agree with your other points, to a lesser or greater extent.
Otherwise, I somewhat agree with your other points, to a lesser or greater extent.