I have the solution, ok?
Kick out Snake and replace him with Alan Rickman.
Because then you wouldn't be able to snap his neck, you'd only be able to snape it.
Which is something I'd very much like to see, namely because I have no idea what that entails.
There, I just solved your entire problem, babies.
Oh a real note, I can somewhat understand the neck breaking to be a little...out of place. Others have pointed out that "well there are swords and bombs and mythical creatures who also have swords in the game, and those would hurt people in real life."
Yeah, but there is some measurable disconnect between the two. I don't think anyone is going to fight a giant monkey in real life, especially not one wearing a tie. There's no blue haired swordsmen running around either (save for cosplay nerds, who do not count), and a wealth of the other things in Smash are clearly in the "cartoony violence" category.
However, a realistic (despite how tight his ass might be) human character snapping another character's neck? That's something you could see in real life and be grossed out by it. I realize the same goes for guns and bombs and such, but this is something you have a higher chance of seeing actually happen, and theoretically could happen almost anywhere and at anytime.
So I can see why it might be a little gruesome, considering the game itself is meant to be so over-the-top with comedic-based slapstick. It's kind of like that art by that one (horribly unsettling) guy that draws Mario with super muscles, gross sweat-ridden hair, and stomping on a Goomba, complete with his innards spilling out. It's that kind of abruptness that sort of throws you into a state of "wtf?" for a second.
This also might hearken back to when the 64 version came out in the USA, and all the "realistic hitting" sounds were replaced with bowling-a-strike sounds. I doubt that something like that is residual enough to remain in the discussion, but I did want to point it out.