So this is because as a quick test version we deleted ALL of the constraints. This is mostly okay as almost all of the parts of the skeletal mesh are all bound to a single bone. The torso (and maybe some other part) has no such luck though. The torso is bound to multiple bones who no longer have a constraint to one another so when they go flying off in different directions they stretch the mesh all over.
For a quick fix you could reset the physics asset and only delete the constraints that work well like this. Leave the torso’s joints constrained so it will fall off nicely.
Your final meshes will need some logic to know which bones to actually allow detaching based on the character type. A fully organic creature with all welded verts will not handle this approach well. You would need to have your binding and vert welding set up to properly allow parts to tear off. Something like a robot would usually be simple to do this way but if you want to bloodily tear off a humanoids arm you may have some issues to deal with.