An ice axe is a mountaineering tool for snow and ice. Designed to brake when your sliding down a slope, chop snow/ice steps, anchoring in snow (deadman anchors or driven in to the snow vertically) and as a third support when you need more support. Yes most climbers will poke the spike in rock too, since most routes are mixed snow/ice/rock.

The basic ones are fine for ice trails like douwalkabout mentioned.

But for scree and dirt? And to hook the adz or pick onto a tree or rock? If you are off the snow/ice and you are not just above a snow/ice slope, it’s time to pack away the ice axe. No reason to use pointy things when not needed. You must realize, hooking an ice tool and pulling yourself up on it, also means you are pulling a heavy pointy thing towards you when it slips.

Straight axes are not that great at hooking anything. The pick on a straight ice axe is for braking in snow. If you want to hook on to rock, you need a different pull and pick angle to prevent it from slipping. Yes, it probably won’t slip if there is a very aggressive angle on the rock, but unless you have a lots if fresh broken off rock, you probably have far less to hook on. So don't use an ice axe for these purposes.