Because it was the analog world and no other (easy) solutions exist. Providing an exhaustive answer will require too much space here, but I'll give the idea with the help of some figures.
So, this is the back of that instrument:
as you can see, there are three contacts. How do they work? Let's check the electronic schematic:
This is a Desynn system. The trasmitter needle rotates over a resistance and changes the amount of current delivered to the receiver (indicator) that rotates depending on the magnetic interactions generated by the three star-connected windings (you can find a good explaination in this video).
And then, let's look at the actual "sensor" (not the best picture):
The round box on the left is the tramsitter, while the bolt on the right is the one attached to the flap surface. As you can see, there are at least two joints between the flap and the trasmitter, and the movement of the trasmitter is no more linear with the angle of the flaps. This explains why the arrow doesn't move linearly, because not even the "sensor" (tramitter) receives a linear input.
The reason of having joints is probably due to space or other mechanical contraints inside the wing. The reason why not to fix this in the electronics is just cost and complexity of doing this in analog electronics of '50s.