From all of the abandoned tree steps I've encountered over the years that are grown into healthy trees I would say the chance of killing a tree with a treehopper drill under normal conditions would be unlikely. Trees receive much more damage from wind and falling trees, than from a hand drill.
I used landscape spikes for years on presets. I would buy them by the case and leave them in the tree. A 3/8" boring bit, the DeWalt and a couple taps from a hammer and I was good to go. Some of those trees were set 20 years ago. On the faster growing trees, once the spike was almost grown into the tree, I'd cut them with a saws-all, redrill and put in more spikes. I have a lot of trees scattered around with at least 20 spikes in each of them. Not a one, shows any sign of damage, all these years later. I realize this isn't exactly like leaving an open wound in the tree, but I beleive its close enough for reference.