Well, I’ve made some progress with this project. After a lot of time in the attic, I have managed to find a way to pull the cable to the box in the living room which currently has 4 unused coaxial ports and an unused POTS port.
Long story short: there are way too many blocks, harnesses and other obstacles in the way of just dropping a fish tape and pulling up a cable. But there is plenty of excess (and unused) coax in the attic. So I can pull one of these cables out through the box in the living room, tape to pull the yarn to it, and pull it up again along with the twine.
Unfortunately, I did not find the engineering yarn I have previously used to pull cable with. So I left one cable separate from the socket, filled everything back into the wall and ordered things to pull the cable with.
Then I did some research for the other end of the move. It would not be good to be able to connect only one end, right?
While the house was being built, I took large pictures almost every day of the building, and they were crucial in identifying the obstacles in the living room wall. They also led me into the big obstacles in the other room ̵