Considering that on our last camping trip 2 weeks ago we were taking inventory of tools to cut wood for our fire. I realized I had 4 axe-type tools on board (and a small bow saw, and a machete). None of them make any obnoxious sounds and the smells that emanate from them (or their use) is purely natural. There's the small Wetterlings under the front seat, a full size single-bit in the topside box, ditto a smaller single-bit, and a Cherokee throwing tomahawk in the camper. Guess you can tell where my opinion lies.
Not much bothers me more than a generator in camp, or a power saw, but I enjoy the sound of wood being chopped. If you wear solid leather boots under your jeans you probably enjoy an axe more, and if camping in Birkenstocks and shorts is your choice, get the bow or folding saw (ok, that's tongue-in-cheek!)
But, and this is important, heed the earlier warning that using an axe is a dangerous sport (as if a chain saw isn't!!). Skill and experience makes the difference between an axe being an efficient tool or a self destruction device.
A bow saw has it's place too. A good 36" bow saw will clear a foot diameter tree from the trail much faster than an axe, and probably safer for most users. A gas chain saw in the same emergency situation makes either manual method seem dumb to even consider.
Fun discussion!
Bob