"Anime" has the great benefit of being quicker to say and type than "Japanese Animation." I'm also not sure how the French version of the word can be pronounced with an accent and the Japanese version without, because without an accent it becomes "ay-nime."
Complaining about the term by claiming that "it should all be cartoons" makes as much sense as complaining that sub-genres in jazz shouldn't exist. On one level, this may be true, but on another level, Louis Armstrong's trumpet playing style is aurally distinct from the bebop trumpet of Dizzy Gillespie or the multiple different styles of Miles Davis (cool jazz, modal jazz, or fusion). I'm not going to ditch any of those terms, no matter how non-sensical they may seem, just because "it should all be jazz." There is a non-trivial semantic loss by doing so.
This is not an invitation to engage in the usual tiresome semantic debate over "what is anime?" by the way

.
The analogy to jazz is also the key to understanding why there isn't a distinction made between the animation of different countries all the time. For that to happen, you have to have 1) a distinct difference between styles, and 2) enough of the work produced in a particular style for a shorthand term to be created for it. I like Canadian animation and there's a lot more of it than you'd suspect, but for the most part I can't tell a Canadian cartoon from an American one. I also can't tell the difference between a lot of French cartoons like the latest
Fantastic Four series. It'll take something like
The Triplets of Belleville for me to see the radical differences in style between an American cartoon and a French one, but it's the only example of a recent exported French 'toon I can think of off the top of my head.
The other major reason why I think Japanese animation gets a special term reserved for it is to dodge the unspoken connotation that "cartoon" (and "animation") has with "for kids." There was very real value in making a distinction between a "cartoon" and "anime" to ensure that some unsuspecting mother didn't buy
Urotsukidoji for her kids. I think there is still very real value in that, because "cartoon" is still "for kids," whether we like it or not.
The Simpsons and
South Park and their ilk are just considered "not cartoons" by the majority of the American populace. I think that's changing, but very slowly.
-- Ed