I enjoy working (carefully) with bladed instruments, and I've even built myself a damascus katana for hard usage (pattern welded, not Japanese layered technique).
I consider the sword the ideal non-firearm weapon when mounted on horseback, which certainly would be a common conveyance in a prolonged SHTF scenario.
But, for taking off with just your BOB, rifle, and sidearm, I think the tomahawk would be as devastating as could be, and certainly more useful than a sword in most cases (simply because you can use the tomahawk to smash open light barriers where a sword would be useless).
A camp axe is ten times better for cutting wood, but too heavy for my BOB (which has a light folding saw too).
I'm reading the history of the colonial "home invasion" experts of their day: Samuel Mason and the Harpe brothers. They committed a good percentage of their murders of armed (but inattentive) frontiersmen with a tomahawk.
For a light to medium heavy pack, on foot, bad things all around you, I want a tomahawk.
I'm in the Florida panhandle, and have a few friends in the 7th Group here. About 1/2 of them carry a tactical tomahawk when deployed.
I bought a CRKT Kangee.
It's a compromise.
They won't let me have grenades.