I think you comments are right on, and absolutely correct. However, I have a lot of experience on the specific trails the OP will be traveling in the Catalinas and Rincons when they were snowed in, and, oddly enough, an ice axe is seldom, if ever, necessary. On the other hand, ski or hiking poles, with baskets, will come in handy. Although they may be superfluous at times on the trail, I'll bet they will be welcome when in deep snow.

The typical trail in those ranges cuts through heavily forested or brushy conditions. Cartwheeling is an unlikely event. Besides, it is strictly against NPS policy guidelines.