I can see few reasons to bring a folder into the woods, and they've all been mentioned already. You'll bring a folder if you're used to always using one, or if you can't legally carry anything else, or if you're just irrationally in love with folders like someone might be with, say, a brand of beer. Or, you'll bring a folder because you feel like it does the job well enough and has the advantage of being compact and light.
I have always said that in most situations, a person can survive without any knife, folding or fixed, and indeed without any tools but his hands and brains. There are some situations where carrying a blade of some sort is a necessity, but these aren't very common at all. I have, for the hell of it, done some long hikes (numerous days in the woods) without any kind of bladed instrument, and they were all very simple affairs - the lack of a blade only caused minor inconvenience, but that's because I knew what I was doing. If I had screwed something up, like gotten lost in the rather vast forests, a blade would have come in more than just handy.
With that said, when I carry a knife, it's going to be a fixed blade. Why? Well, why not? Why not carry the kind of knife that is both more reliable and more ergonomic in the hand than any other? Why should I carry a folder, a knife already broken in half and by definition unreliable, if I can carry a knife that has very simple construction and is much more reliable and even feels better in the hand? Why should I carry a folder when I can have a more effective cutting tool? I see no reason. Being that I'm not a child, the weight of a small to medium fixed blade is meaningless to me. A 3" to 4" fixed blade is very light indeed to carry, and the weight makes no difference to me. As for the length of the knife, that matters just as little. I carry a rather large backpack, and if I can carry that, I can certainly carry a couple of inches of knife with me. In a good sheath, a fixed blade is a breeze to carry.
There are endless reasons why the fixed blade is superior. Obviously, it is far stronger. It is simpler, far more reliable, and far safer to use. A fixed blade will never, ever close on your fingers, no matter what the hell you do with it. For anyone worried about having their fixed blade run through its sheath to stab them, I suggest you get a better sheath with a strong inner liner. With a good sheath, a fixed blade will not be coming through it, and if it does, the impact is going to be such that it would likely tear apart any folder's construction, too, and have the folder's blade stab you in the guts. The fixed blade is ready to use once you draw it. You never have to play around with uncertain one-handed opener features or fiddly automatics that might fail because there's snow and sand in the system. A fixed blade is simpler to clean, compared to any folder. But best of all, the handle ergonomics on a properly made fixed blade are superior to a folder's, because the handle can be designed only with ergonomics in mind, and it does not have to be large enough to accomodate the blade of the knife inside it.
I understand that some people will not carry fixed blades because of legal reasons. I'm fortunate enough to not have that issue: where I live, a folder is exactly as illegal to carry as a fixed blade, unless you have a good reason for it such as being out in the woods hunting, in which case you can bring a two-handed sword for all anyone cares.
Finally, there's tradition. To call any folder traditional is, to an extent, an insult to tradition. Does anyone here think folding knives were around four thousand years ago? Three thousand? Romans may have used primitive folders for purposes of providing handy eating utensils on the field, but they would rely on fixed blades for any serious use. Traditional knives are fixed blades, tried and true. They worked from the beginning of the ages, and they work still. They work better than anything else. Sure, grandpas in some places of the world used folders a hundred years ago. In some places, they used fixed blades instead. Such is life. Sure, you can call the old folder that your grandfather used traditional, because it's a hundred years old and has wooden scales and a carbon steel blade. Sure, it's much more traditional than flashy modern mall ninja folders with serrations and one-handed opening features. Sure, it's a nice knife, and may be worth a lot of money. But when we start looking at what people used a really long time ago, and what still survives to this day, that would be the fixed blade.
Well, that's what I think, anyway. I've never been one to like folders. My father used fixed blades and only fixed blades, and so did his father and his grandfather. Why use something that's only decent when you could use something that's good and even great?
But Skammer as Noss has pretty much proven there is no such thing as an unbreakable fixed blade, any knife will break if you heap enough abuse upon it!
Certainly there is nothing unbreakable in the whole world of man-made tools. But, you mention Noss' testing. Look at some of the things he does. You will find that some fixed blades will think nothing of being hammered with a steel mallet through 4x4s or having a large man stand on their handle while the blade tip is wedged tightly in wood. If that's not unbreakable enough, then I'm thinking it's best to stay indoors instead of braving the dangers of the outside world. Sure, you can break the strongest fixed blade - unfortunately for this argument, it will require you to try to baton it through steel with a heavy hammer. I don't know about you, but I've never had to baton my knives through steel in the woods. I can't think of any scenario where I would need to, either. And if I did, I guess a folder wouldn't be very useful in that scenario, would it?
