So, unlike with knives, there is a general agreement that the edge ( I don't mean cross-sections of the bit) should be convex?
There's no big mystery with bevels, as they should line up with eye. So you put some tape over eye area and file or grind away.
Once you get close to filing to edge, you can switch to drywall sander and emery paper and slightly convex. That means that the whole bevel will be slightly convex, with the convex steepening at the edge to give it more robustness.
The simple advantage of having some convex is that once the edge bites into the wood, the chip is wedged out by the bevels without the edge actually touching much wood. That causes the axe to keep it's sharpness through a lot of chopping.
Some axes have very pronounced convex cutting bevels or very wide gently convexed bevels. This causes big chips to go flying out of the cut, and so keeps everything clear for further chops. You sure don't want to be stopping after every chop to pick out chips which have fallen back into the notch, and you can't count on always being able to chop in from the sides, so that chips fall out.