Why do you need an ax or hatchet? Unless you are 'creating a situation that requires it' for the express purpose of playing Mr. Bushcrafteer and destroying some part of nature?
This whole bushcraft crap has pissed me off many times in the past when I went hiking, and came on a clearing that had been created by some energetic numbskull with his big chopper and hacked down everything he could to prove how good his big blade was at destroying nature. The area littered with remains of small trees that had been hacked down, then into small pieces of debris for what purpose? So some idiot with more money than brains could play bushman.
For a few dollars, you can drop a plastic 10 foot by 10 foot disposable drop cloth from the paint department at Walmart or Home Depot. Its about the size of a paperback novel, weighs about the same, yet can be strung up in a few minutes making a totally waterproof windproof shelter that is tons better than anything you can make in 2 hours hacking away at the wilderness with some big knife or hatchet. I know, because I've done it.
When I got out of the army with my right foot and ankle pinned back together and had to go to ultra light backpacking, I got a light nylon waterproof tarp that rolled up about the size of a loaf of bread. Very light weight. It gave me a 12 foot by 12 foot area that I sat out thunderstorms under staying totally dry. A ping pong ball and a roll of jute twine completed the shelter. Never had to cut so much as a twig. Put the one ping pong ball in the center of it to hang it up like a big pyrimid for shelter. Ran some twine out to saplings to the sides to make a waterproof tarp tent. A coffee can hobo stove used small twigs for fuel so no chopping needed. If it was wet, some army heat tabs were used to boil water for coffee or oatmeal. Once I sold off that Ridiculous Randall 14 that I never used for anything, I carried my Wenger SI to deal with any cutting or cooking job, or slicing up a nice Gouda cheese for trailside snacking.
All I ever saw from the bushcraft idiots was areas of nature devastated by irresponsible hacking away at the woods to prove what a great thing their big chopper was.