You're in the minority. I don't think I ever heard anyone complain about the team name. Best team name in the league, imho. Glad we didn't get a stupid made up name like the New York City Straphangers.
I've almost never made an issue of it because how many things am I going to complain about and I decided to just make my peace with this, but since you brought it up...
1. We're not in Europe and don't name teams this way.
2. Especially pointless since it's not an actual club created from the ground up by volunteers and community leaders. I particularly don't like adopting a convention built on a tradition while ignoring and rejecting the actual tradition.
3. We don't call it football here. The New York football club would be the Giants, except they're in NJ. As are the Jets.
4. Seven syllable name.
5. Five-syllable acronym.
6. No pleasant rhythm to either one.
7. Despite the ungainliness, there is still no consensus what to call the team for short. Which should have started somehow by now. But all the likely candidates have substantial factions that hate them.
8. Because the most likely shorthand for New York City Football Club, is "City," and substantial numbers of folks on both sides of the Atlantic would find that annoying.
9. You can call them "New York" but every team in NY is already called that in context; this team needs a unique ID.
10. New York City belongs to the city, not a team. Unlike MC, for example, where nobody calls the city Manchester City.
11. Same for NYC.
12. I say the name to friends and family who haven't heard of the team and they generally think it's the dumbest, most boring, long-winded name they've ever heard. Sometimes I wonder why they didn't put the word "Generic" in front of it. And yes most of these people are not the target, but it would be nice not to have to digress with an explanation about the name whenever I mention going to a game, for example.
And I could go on.
But now I've gotten that out of my system I'm not going to argue about it. It's not going to change and I'd rather focus on other stuff. And if you like it -- good! From what I can tell I
AM in a minority in that MLS seems to have determined that these sort of names work.
*
Finally, as shorthand I'm partial to CFC even though it has many of the above flaws. Sometimes logic doesn't rule. I just think it has the best rhythm of the alternatives.
* I will however always pronounce RSL as "REEL" Salt Lake just to be contrary.