...It means that range of linear velocity of blades is from 2v to 0. This situation will destabilize blades and eventually break apart. To begin with, blades will not rotate at all. But as we know, this does not happen.

Blades will not break apart if one part moves with zero velocity and the other with the velocity 2v. It is not VELOCITY which breaks blades, but FORCE which is mass times ACCELERATION: F=ma. Acceleration does not change if you add constant velocity v0. Indeed, in moving with v0 system new velocity is v'=v+v0, so dv'/dt = dv/dt + 0, or a'=a.

Same as tires of your car do not fall apart due to the fact that lower part moves with zero velocity and upper part with twice the velocity of the car.