There are plenty of walking routes where having a wooden pole dangling would be more than a bit silly. Crib Goch in Wales comes to mind.
I did say it depends upon the difficulty of the terrain, notwithstanding the first ascent of the matterhorn was carried out with the assistance of wooden alpenstocks. Crib Goch is not something I would consider these days, I suffer from vertigo but last year I did go up and down the Watkin path with a hiking staff in one hand and a walking stick in the other feeling somewhat more secure than if I had only feet to rely on.