I think that the most important aspect of the axe with regard to carving is the bevel shape. I've had a few hatchets, from cheap to GB, and most were verging on dangerous when carving. The bevels were strongly convex, which is a good thing if you are splitting wood (helps prevent the head from sticking). However if you are trying to take shavings off, the rounded convex bevel is prone to making the hatchet glance off the work sideways.
Convert that convex bevel to a chisel (as in a side axe) or plain flat bevel (like in a scandi-grind knife) and it is a different story. The axe bites in and can also be used like a plane, shaving wood off.
A curved edge takes this a step further, allowing you to 'slice' shavings out, turning the axe head out of the work.