Problem with moving origins of linked objects

I have a few objects that share linked data. I want to adjust position of their origins, so that each of them has its origin at bottom centre. So I select them all and go to Transform > Affect Only > Origins. Now I can move origins using gizmo, and it behaves how I want - only origins are moved, and geometry doesn’t change its physical position relative to world. But I can’t do that using G key, because geometries will move too. I don’t know what I’m doing wrong. It used to work. I swear I could use G, and it used to move origins only.
I’ll provide a Blend in a minute.

I choose transform orientation to Local, and that lets me move origins only with gizmo, but I still can’t move them with G key.

Here’s the model:
origins.blend (1000.9 KB)