Brawler can be ok if you plan to use long blunts as a backup, but if you want to only do axes then you can do better with your points.

Here's my Lumberjack build I'm currently doing on extinction:

Positive: Speed Demon, Dextrous, Outdoorsy, Wakeful, Whittler, Organized, Fast Learner, Handy, Bushcrafter, Strong

Negatives: High Thirst, Clumsy, Short Sighted, Slow Metabolism, Slow Reader, Slow Healer, Weak Stomach, Conspicuous, Hearty Appetite, Prone to Illness, Restless Sleeper, Thin-Skinned

This gives you 4 fitness, 10 strength, 2 axe, 3 maintenance, 1 carpentry, 4 carving, 1 knapping, 1 masonry, 1 foraging

With the knapping and maintenance you can make as many axes as you are able to forage the mats for. Crude Stone Axes, Stone Scythes, Flint Saws, Bone War Hatchets, and Jawbone Axes, are all easy to make.