OK, I confess, I'm not sure what sort of "non-extreme" conditions those would be. If the slope is steeply inclined, as you describe, then it's no longer non-extreme. If you slip, it's extreme. Trekking poles help you keep your balance on slippery terrain, but as a self-arrest tool, they're as useful as a dessertspoon. Denali's West Buttress route has a long, moderately steep stretch of ice and snow that rises from the 17,000-foot camp to Denali Pass. It looks utterly benign, and climbers often attempt it with trekking poles. But falls are common on the descent, and fatalities usually follow if the pole-bearing climber can't stop him or herself before tumbling down the slope.
However, just buying an ice axe doesn't solve the problem, either. Ice axes themselves are pretty hazardous gadgets, with three pointy ends that can impale a foot, calf, or chest wall. So while I recommend you get one, I also recommend you find a way to get proper training in ice axe and self-belay techniques. Otherwise, you'll be carrying this thing, will actually NEED it, and will be thumbing through the instructions while speeding down a snowy slope.