When you repack for a SkeletonGraphic, you need to update the texture it uses by calling:
skeletonGraphic.OverrideTexture = runtimeAtlasTexture;
You can see this in use in the MixAndMatchGraphic example.
In the case where you didn't repack: The limitation of SkeletonGraphic is that it uses Unity's MaskableGraphic class which is only able to use one texture. If your skeleton uses multiple textures, it always only use the first texture for all parts which always results in incorrect-looking skeletons.