As of 2.79, how stable is the skin modifier?

That's odd. I've never see the skin modifier glitch the way you describe - not for as long as I've used Blender. So it should be plenty stable.

Oh wait, I got confused - thinking "skin modifier" was the "armature" modifier. The skin modifier is still finicky at times, but usually with enough patience you can tweak it to be what you need. That's been my experience. While it can be annoying when it twists your mesh up or creates a whole, at the end of the process a basemesh doesn't need to be perfect. You may need to alter the position of vertices away from ideal placement to avoid holes, but that's ok. You'll sculpt it into proper form later.