The wire is anchored on the right side, and guided on the left. This lets the left end of the wire project thru the tang hole, and keeps it from moving back and forth. As the wheel rotates, the wire pushes the gear around ahead of it.
Thus the push is in line with the plane of the gear. There is no need to have the wire anchored on the left side. As long as it projects through the tang hole it will push the gear.
Anyone can try what you suggest. In fact I have done so, and believe that what I have is better, as it provides discoupling when resistance is encountered, keeping things from getting bent.