I think that countable vs. mass objects is a miss-conception, and that it is better to think in terms of fusion. The real distinction is between objects that are assemblies of other objects, and objects that are fusions of other objects. For a fusion, the whole is made up of assembled objects and is precisely the sum of the parts. Now assembly can happen at a number of different levels, e.g. atoms to molecules, molecules to simple objects, simple objects to complex assemblies. Similarly fusions can happen to different levels of assembled objects, so a plasma, a piece of steel, a car collection.
Countable objects are clearly assembled objects, and mass objects are clearly fusions.
You would normally think of screws as countable objects, but it is common to sell them by weight. This is clearly a fusion, but is not so clearly either a mass or countable object. The fundamental issue of how to model these concepts remains. No, there were things that were neither mass thing nor count thing, or maybe both, because it was unclear what the basis was for the mass and count thing terms. I'd like to learn, but I don't understand yet.