I don't know how practical the walking stick/axe is as a tool. There are others with much more outdoor experience than I do here. But for defensive purposes? Well, the axe and axe-like polearms were great at opening cans of knights on a medieval battlefield, but not so great in a civilian context. The axe is heavy and slow. Recovery after an unsuccessful strike takes a relatively long time, and that leaves you open to a counterattack. It takes a lot of energy to wield, which is never good news. (Who'd want to carry something like that on an arduous hike?) Though it's powerful, someone with a lighter, more agile weapon with the same reach will be at an advantage.

Look at how many schools of stickfighting, swordsmanship, etc. survived all over the world. Then take a look at the schools of axe fighting. Right.... The weapon was unpopular for a reason: except under special circumstances in which it excelled, it sucked. Most people drifted to superior all-around weapons in the civilian context, and so it made sense to formalize techniques into curriculums and schools. There was a market, there were students. Axes? They were for the unusually strong or the idiosyncratic folks.

I have no experience fighting bears, wolves, or zombies.