The flip-side is that a closed set of movement related words uses a different auxhiliary than the rest of the verbs, in German as well as e.g. French. However i

The flip-side is that a closed set of movement related words uses a different auxhiliary than the rest of the verbs, in German as well as e.g. French. However is gone, did go, has gone, was gone do not fit the pattern. Deriving the reason for the different helper verbs from first principles is endearing, in principle, but you've merely given names to it that you somehow take for granted. It makes sense to have cut wood, and it makes no sense to do come or have come, except for a potatoe plant which if it cometh will have a Keim or Keimblatt to be precise; I'm guessing here. German rather differentiates is and have in the perfective, "sie ist ge-kommen", "sie hat ge-sehen"; We don't say has come because we are intuited to think it is wrong. That is, you might as sell say the crisis is not ye(t)come.