It does! The force creates a translational acceleration of the center of mass.Another question I would like to ask is, consider the free body diagram there is a translational force in the direction of the smack, why isn't the whole stick move translationally according to the force provided?
No. The same force does two things: It accelerates the center of mass and it creates a rotational acceleration about the center of mass.From your example it seems that there is a portion of force used to cause rotational motion and the other portion causes translational motion.
The entire force is used for both. (Of course the amount of torque produced depends on the point of application.)What factors determine the the fraction of force that is used to move the stick translationally and that used to cause the torque?