The same could be said of Doctor Who and how funny it is that all the aliens and all the humans at various future points in time are British. Or how all the Imperials in Star Wars tended to be British, too.
First of all the first HMS Enterprise was a captured French warship with the French equivalent of Enterprise. The First America Enterprise was a later HMS Enterprise that we captured. Just so that we all have that straight. Plus think the OP is missing how diverse the crews were, even back in TOS, and during Enterprise. It's always been a mix of different races of humainty and included alien officers in Department head rolls. It's not like in the live action "Space Battleship Yamato" where the Yamato is sent off to save Earth with the best of Humanity, and the all apparnetly happen to be japanese. :p
well it is an american tv show made by an american, using the English language. nothing anyone can do about it, thats just what it is.
the program itself hardly deals with america at all though. the original crew has people from africa, japan, scotland, russia, america, and vulcan. The TNG cast was equally as diverse.
You could probably argue that there are more american characters than from another other county, and that may or may not be because its filmed in america or because it has a largely american audience but that would be nitpicking.
As said above the enterprise has roots in other counties as well. so im not seeing it.