Helicopter flight: Does the rotor disc plane and hub plane rotate along axis perpendicular to rotational axis?
The shaft does not tilt, but the disk does.

You understand the function of cyclic control correctly. It indeed changes the pitch of the individual blades, shifting the centre of lift.

However, the blades are not strong enough to take that much bending load. If the rotor was completely stiff, they'd quickly break off. So instead the blades are allowed to flap up and down. That allows the plane of rotation of the blades to change first when you apply cyclic input and the body attitude follows.

The body attitude follows. When the rotor tilts, the action like of lift gets tilted with it, so it no longer passes through the centre of gravity, which will create a moment that will try to pull the body into the attitude corresponding to the rotor. With individual flapping hinges or flexing there is additional, more direct torque on the rotor hub trying to do the same, while on teetering rotor this is the only effect and the way cyclic is rigged prevents the rotor from trying to tilt too far relative to the body.

See also How are pitching and rolling moments transferred from rotor to the body?