The SkeletonRenderer component basically generates a Mesh at a MeshRenderer every frame, with Materials being automatically assigned every frame according to which atlas texture is currently needed. Unfortunately I'm not aware what the bink video scripts do in regards to the mesh at a MeshRenderer, but it's more likely that the problem is based on the skeleton's Material replacing a bink specific Material. You should easily see which materials are assigned at any time using the Inspector.
See here on materials in spine-unity
spine-unity Runtime Documentation: Materials
You might also want to have a look at the OnMeshAndMaterialsUpdated event callback, if you need to run custom code after the skeleton Materials have been assigned each frame, and SkeletonRendererCustomMaterials is not an option for you: spine-unity Runtime Documentation: Life cycle
