The OP asked about ADK peaks. I did a lot of the ADK peaks in winter (including Gothics) with one ski pole and old bearpaw snowshoes, occasionally crampons with no front points. Depends on conditions, YMMV.
One bad thing about poles: you can't self-belay with any confidence. I use a whippet (ice axe head on a ski pole); I take off the basket to attempt self-belay, and there are still times when it hangs up at the first joint. I've bent the damn thing several times.
If he wants to ice climb at Chapel Pond or Cascade, that's a whole different animal. Get separate mondo ice climbing tools. That's not my bag, but I never saw people up there with mountain axes.