I have used several of the pocket chain saws and they generally perform well. Some cut faster than others. They do have a tendency to break if the motion of the cutting is not kept straight. Any angle introduced during strenuous cutting can break the links.

Wire saws do not cut as well, but will work. Any binding on the wire blade will lock it up tight. It is best to push on the item being cut so as to open up the gap a little and allow the saw to move freely against the surface being cut.

There are many folding saws that a re a little bigger than what I would call pocket size (depends on the depth of your pockets). Most will cut wells, but as Hikermor says, there are some durability issues.

For larger logs a saw will cut efficiently with less effort than an axe. Small saws are good for notching wood for construction to strengthen your lashings. Saws are less efficient for lopping off smaller branches, skinning animals, scaling fish, carving or sharpening wood to make tools, cutting bamboo (yes we have it here, and a type of it is native). You need the right tool for the job at hand, and if you don't take it with you it will not be at hand. Better to have and not need than to need and not have.

Doug's opinion is a well informed and experienced one. It does not replace personal experience. You can read all of the blogs, watch all of the Youtube videos, read all of the books on survival, but you need to get out there and learn how to do it yourself. Nothing develops a skill better than practicing it before you get into an emergency situation.