While you should know how to self arrest properly with an axe they have many more uses than self arrest. This is just a quick overview of their uses I copied and pasted.
■When you walk on snowy steep ground, an Ice Axe will provide extra stability and will also help prevent slipping. It will then act as your third leg.
■When you accidentally slip down a slope, you will be able to use the Ice Axes as self belay if you push the shaft down into the snow.
■If the slip turns into a slide then you can use the Axes as brakes. The Axe is then held across the body and the pick pushed into the snow.
■The Ice Axes can also be used to cut steps in hard snow when Crampons aren't being worn. This is particularly useful if you are just crossing a short section of snow and do not want to put your Crampons on.
■To dig seats or large steps (for resting, organizing equipment, etc.), pits for checking snow profiles, or snow holes, the Axes can be used as a kind of a shovel.
■When you use the Ice Axes together with a rope, then they can provide extra security on steep snow slopes to construct bucket seats, snow bollards, and buried Axes.
■You can use the Ice Axes while you are climbing on ice, hard snow, frozen turf, or rock. The pick is then used for hook onto bulges or steps, swung into ice, or twisted into cracks. The Adze, head, and shaft may also be jammed or twisted into cracks.
Personally I often use it for the first thing although most of the time I do it probably isn't 100% necessary. The four times I can think of that I would have actually been uncomfortable without it and turned around were on the snowfields on Jefferson last March, Monroe's summit cone three years ago (also in March), The Franconia Ridge in March a couple years ago amd the Lions Head trail a couple years ago (almost in March, Feb 28th to be exact). Seems to be a march thing around here!
Occasionally I have used it to cut steps for people without full crampons and used it quite a bit on the Star Lake trail last winter to grab/pick onto rocks on the Star Lake trail which was a mess of mixed ice and rock.
Never had to self arrest but have practiced a number of times everywhere from Snowbanks, to Pawtuckaway, to Washington. Fun to practice and do, at least when your life isn't on the line.
Also had to use it once in a rescue to lower someone down the upper section of the Ammo Ravine trail. Drove it in and lowered them down the steep sections by wrapping the rope around it and letting it out slowly. Full crampons helped get traction for that too but what are the odds you'll ever need to dig into ice really well for something like that? Probably and hopefully a once in a lifetime deal.
Obviously others are fine without one. Personally I prefer to have it for the rare occasions its truely necessary and for the extremely rare case it's needed in an emergency situation. Really not that heavy to strap on your pack just in case IMO.