Well. . .
Like I've mentioned before, if you look at history, all cultures had a big/small duo. Some sort of big chopper and a small knife. Even Nessmuk used his hatchet and pen knife for the vast majority of his outdoors work (by his own admission), and the "Nessmuk knife" got comparatively little, specialized use.
I think it's well known that in northern climates the axe is king. The most popular was the "trapper's axe". About 24" haft and 1.5-2 pound head, depending on the maker. These really are not heavy, and for their versatility, well worth the weight. Most of the weight is in the head, so a 24" axe doesn't really weigh that much more than say, a 15" hatchet with a comparable head. For clarity, I define hatchet as a tool designed for use primarily one handed, while an axe is designed to be used primarily two handed.
Axes are only dangerous when used improperly. If one is not going to put in the time to learn to properly use one, then sticking with a saw is probably a better idea. Not picking on anyone, just stating in general. Usualyl axe accidents happen because people:
1.) Don't keep them sharp
2.) Don't watch where the follow through/miss will take the bit, and plan accordingly
3.) Don't understand the proper geometries to use when using an axe (a good example is the Youtube video of the guy who's saying that the Fiskars axe suck because beating away with almost straight on blows with a SPLITTING AXE didn't work well.)
The axe is very versatile and very effective when properly used. You have to decide if it's worth developing the skills. Geeking and flailing with an axe
is dangerous.
Large knives (also machetes, kukris, etc), are a little safer than the axe in unskilled hands. Generally, a miss with the sweet spot on the blade will yield a poor hit either tipward or scaleward, depending on if you misjudge long or short. So you are more likely to bury the blade in the wood, than in your body on a "miss". Some people just like blades better than axes. Machetes are also very light for their size.
In reality, if kept properly sharp, the chopping implement is going to be faster than the saw (unless it's a powered saw). A wild swing with the chopper is more dangerous than blade breakage on a saw (seen it happen, it was bloody, but no loss of limb). The only real advantages of the saw are safety and lighter weight. Each person has to pick the optimization that works for them.
Now, the neck knife. It depends. Are you restricting this to "neckers"? In which case, I hate them.
However, if you are including knives the size of an RC3/Busse Active Duty, they are wonderful companions to the big chopper. Also, I tend to carry my choppers on a baldric (except axes/hatchets), and the smaller knife on the baldric strap. I've posted pictures of my kukri/R10 rig. So there's a 5 inch blade, carried as easily as a necker, and just as accessible and unobtrusive.
As for the question of why chop when you can find smaller deadfall, or cut down a sapling, well. . .
1.) You may not want to fell a live sapling.
2.) There may not be any small deadfall. Especially in areas that are frequently camped, the small stuff might have been used up by guys who only carry pruning saws and can't process larger trunks

. A lot of the times the trunks I'm talking about are 18-24" in diameter, not something you are going to progressively push into your fire. However, with a proper chopper, you can split off portions which then get shaved for tinder, processed into smaller and larger pieces for the sustained fire. Why not use it instead of felling a live sapling?