...animations work fine... So, I'm relying on your advice, because I really don't know how to fix it..
I really don't understand the issue. If the animations are working, then your model is perfectly fine.
The bone a mesh vertex is tied to is directly related to what MESH OBJECT it belongs to. Remember that objects are also bones when exported. If you make a model up of multiple meshes, then you will see each part assigned to a separate bone. In either event, it's irrelevant as your skin weights override the weighting.
Hopefully this helps you understand how it works. There is nothing going wrong when you only have the root mesh bone assigned to all the vertices. It's not something that is "right" or "wrong".