"I must be missing something… Why do you need a saw/hatchet again? Are you camping in the middle of winter and cutting down your firewood? I'm perplexed."
This place we go, the 4 of us, is 3 miles in from the road and hosts backpackers and mountain bike campers, mostly backpackers though. It is very popular, so by August when we go, there is little in the way of small branches, 1-3" in diameter on the ground. There was however, a huge hurricane that blew in a few years back. Remember Irene? It took down a huge amount of medium sized trees, say 4-6" and that's just about all that's left. We camp in place for two – three nights with small day trips and enjoy a nice fire in the evenings. This place doesn't get a ton of sun during the day so much of the stuff on the ground is damp.
Why would you backpack to somewhere so popular, well used? It was the first place we brought our kids to try backpacking, they loved it and it's a tradition that is easy to keep. There are a couple convenience factors we couldn't pass up when introducing them to the wonderful world of backpacking, the pit toilet, and the bear box, proximity to a cool natural attraction, and proximity to extremely clear cold water.
"Where are you getting your firewood? Seems like the only time you'd need an ax or a saw would be to actually cut down and process an otherwise live tree, which is a terrible source of firewood for anything other than homesteading, where you'd actually be stacking quartered wood to dry and use for the winter."
I never EVER cut down green wood. Ever, period. I camped in Baxter State Park every single summer until I was an adult. Cutting down green was instilled in me very early as a cardinal sin in the backcountry.