A short alpine axe is weight in the pack for everything except negotiating a steep slope.
A long axe is a cane on the flat, an unbendable/unbreakable assist on a moraine, a step-cutter on the short ice slope when you don't want to stop and dismount to put on crampons, and an arresting tool on terrain where a fast slide/fall/tumble means debilitating injury or death. It's a compromise versus a short axe on the steep - the long axe will fatigue the uphill arm a bit, but we're not scaling K2 here. I'm not willing to carry a short axe.
Dragging my sorry behind to the airstrip with two poles broke off shorter than long axe length colored my opinion of walking aids in glacier country. The poles were great. Until they broke. Then the aluminum stubs slipped off every rock and generally sucked.
This year I carried a pole and the axe. I used both on the moraine at the same time, and just the axe on the steeps where you'd want the ability to arrest. Worked like a freaking champ.
The applications for trekking poles and ice axes in my opinion are quite different with the ice axe being a very specialized piece of equipment required mostly in extreme alpine/high elevation environments. An ice axe is a tool that is a bit more applicable in situations where ice, glacier crossings, extremely steep/slick rock, or self-arrest potential are likely. The trekking pole on the other hand is more of a mobility tool where steep and uneven terrain, long mileage and heavy loads are likely to be encountered. Having said that, there are some instances where an ice axe can be used instead of trekking pole(s) but those in my experience are few and far between. They are two different tools for two different jobs. Now before someone gets offend and flames me, I've used both extensively so I'm qualified to make the observations I'm making having used both over many years.
