>"but it is not supposed to be a walking stick."
I differ on that. On vertical ice, you want a shorter ax because you're swinging it like a hammer to plant the pick and/or drive pro and/or chop away rotten ice/snow before placing ice screws.
But I don't do vertical ice. I sometimes am on high-angle snow and when that gets to be wind-blown or freeze-thawed into ice, I want to be able to self arrest. That means an ice axe. But except for actually taking a fall and needing to self-arrest, I do use it (not as a hiking pole, but) as a third leg of sorts. It helps stabilize me on a traverse of high-angle snow. It can be stabbed into the snow to be an anchor. It can be an additional support on a stream crossing.
And when self-arresting (and all the self-arrest practice you should do BEFORE you need to do it for real): I like a longer ice ax for that as well. The point of a 45- or 50-cm ax rests on my chest and is harder to control than a longer ax (80-90 cm) on which my lower hand has more leverage and the point is well away from my ribs (I hate punctured lungs) and belly (hari kari should be done with a different implement, in a different setting, and only for good reasons).
HYOH
CYOIA (carry your own ice ax)
Those are reasons I opt for a longer ice ax for the mostly non-technical snow travel I do.