It's not on the way to being TN-C-S, it is TN-C-S, because your installation is earthed to the incoming neutral. The fact that it's also connected to the sheath doesn't change that. TN-S means that the neutral and protective earth paths are completely separate all the way back to the transformer; if they share a conductor at any point, then it can't be TN-S, by definition.
That's why I also cautioned about the way in which one can find services which at the house might appear to be TN-S, and in fact which might still look exactly the way they did when they really were TN-S, but which due to changes in the local distribution network are no longer so.
There is bound to be some current flowing if you have any other connection to earth on your installation, and especially if both you and your neighbors still have solid metallic water services which are all solidly connected together.
Both the cable and the head are right to be original for that era.
For the wiring within you house? A local electrician.