balaton,
You hit on a major myth that training books leave entirely unexplored - how the rotor actually controls the helicopter. Most rotors are not hinged exactly at the mast like the simple illustrations in the training manuals, the rotor blades are usually attached to extended arms that are strongly attached to the mast. The arms allow strong bending forces to be transmitted from the blades to the mast, and then to the body of the helicopter, so the body must follow the rotor.
Here is a teetering rotor that is more rare, but often used. With this, only the lift of the rotor can control the helicopter, because the tilting of the lift imparts a control around the center of gravity of the machine.:
Here is the more common hinged rotor (articulated), where the tilting of the disk makes the blades use their centrifugal force to twist the rotor and force the body to follow. This rotor can control the helicopter even if there is little or no lift being applied. That is why it is more modern, and now more often used.
I hate to be rough, but most of the explanations below aren't correct. For example, if you suspend a pop bottle from a string through a loop, at the center of the cap, that most nearly approximates a teetering hinge, and in that case no angle of the string can make the bottle tilt at all.