Ah, the crux of the issue! If you haven't spent time practicing, don't waste your time and energy carrying one. Having an ice ax
does not make you safer than not having one. Having one
AND knowing how to use it,
AND having
practiced with it substantially can save your life in steep alpine terrain.
I would argue that anyone suggesting that a trekking pole has any level of usefulness in self-arresting has probably never taking a sliding fall and tried to self arrest with a trekking pole. If conditions are mild enough that you can self arrest with a trekking pole, you probably don't need to self arrest in the first place.
When traveling on steep ice and snow:
1) Trekking poles and micro-spikes provide a tremendous amount of stability, significantly reducing the likelihood of a fall. And,
if you don't fall, you don't need an ice ax.
2) If you get yourself into a situation where you do fall, and it is steep and/or icy enough that you can't control your decent with your feet/heals
nothing short of an ice ax in practiced hands is going to do you much good!!!
3) An ice ax
used correctly is more secure on steep snow and ice than trekking poles.
4) Using an ice ax is much slower than using trekking poles if trekking poles provide adequate stability.
I've never carried an ice ax while backpacking, and I have not yet encountered a trail condition where I felt I needed more than my micro-spikes and trekking poles. However, I am also a lot cockier and more "sure footed", and more comfortable in extreme terrain than most other people I backpack with. The confidence of a well placed ice ax shaft into an ice/snow field as you walk along an extremely exposed section of trail may enable you to keep going when otherwise you may not.
YMMV
P.S. Ice axes do look very cool hanging on the back of a backpack. So, maybe that is reason enough to carry one.