Practically speaking: The string binds the stone to the bough, and a bough is used instead of a stick because obviously, a stick (for example, 30cm in length and 1 to 2 cm in diameter) would make a terrible hatchet. It's not unreasonable to require two string for crafting since this would press new players towards initially putting a couple points into exploration to find spindly taproots.
The metal axe (hatchet) can remain crafted with only the two metal bars and a block of wood, since the metal can be formed to attach permanently to the handle.
Add a stone and metal felling axe. Currently, the only axes are technically hatchets, pictured below.

For cutting down trees, felling axes make much more sense, pictured below.

The stone felling axe is made with one stone, 3 string, and two boughs. It's 1x3 in the inventory and is held with both hands. It somewhat decreases stamina loss when cutting down trees and can possibly be used to make wood blocks. The idea with the stone felling axe is to not devalue the stone hatchet by being so much better, so the stamina use reduction would be minor.
The metal axe is crafted with three hard metal bars, two blocks of wood, and maybe 2 or 3 leather as a requirement (for an edge guard or grip). This would be an endgame tool for carpenters and woodsmen, with well reduced stamina loss and when felling trees and making wood blocks.
The metal felling axe would also make a decent weapon, and would be a good intermediate between a metal hatchet and a B12.
Of course, the trade off of using these is that they are held in both hands and require a bit more resources.