Well I downloaded your mesh and the issue seems to just be vertex groups. There was no weight paint on the mesh as far as I could tell, could just be the download. When its importing your mesh and trying to force the skeleton on it, it doesn’t know what bone is supposed to control which vertices. Again I’m not sure if it was the download, your mesh didn’t have any skeleton. Another issue that I’ve run into is a mesh that is comprised of more than one scene OBJ on export. For instance you have the eyes and the head and the body and whatnot separated, I’d recommend keeping them separate until export, for easy modifying, then combining all before exporting manually, not during.
My issue came when I tried to keep the eyes of my characters separated for ease, every time on export it would only get the armatures for the eyes. I think it could just have something to do with combining armature hierarchies on export, but I find pretty much a 100% success rate when all scene OBJs are combined and the armatures are combined and parented through one contiguous mesh, just keep the elements you want separated by using separate materials for easy selection and texturing. I would just build a skeleton from scratch and keep a blank for when you need it.