There is an incorrect assumption here:
Assume we have a ball, and we lift it up by a distance $h$. Then, as I understand, the force $F_l$, must have been more than the gravity force, $F_g$. Other than this, the ball would not have moved.
Forces are the cause of acceleration, not motion. Remember Newton's First Law: in an inertial frame, an object not acted on by forces will continue to move at the same velocity. So an object that has a nonzero initial velocity, where $F_l=F_g$, will continue to move at that velocity, and, if that velocity is directed upward, after a certain amount of time will reach a height $h$.
At this point, you can say that the lifting force did an amount of work equal to $F_l h$ on the object; it increased its potential energy by $F_l h$, and its kinetic energy did not change (it's still moving at the same speed). You can also say that the gravitational force did work $-F_g h$ on the object, since the force was oppositely directed to the motion. Since $F_l=F_g$, the net work on the object is zero, which is consistent with the work-energy theorem:
The total work done on an object is equal to the change in kinetic energy of that object.
Let's assume now that $F_l>F_g$. Now the object accelerates upwards; its kinetic energy is higher at height $h$ than it was originally. How much kinetic energy did it gain? Well, the work-energy theorem tells us: the amount of kinetic energy it gains is precisely equal to the net work on the object, which is $(F_l-F_g)h$. Indeed, this makes all of the energy totals add up correctly; the lifting force inputs a total energy of $F_lh$ into the gravitational system. Of this energy, an amount $F_gh$ went to the potential energy of the object, and an amount $(F_l-F_g)h$ went to increasing the kinetic energy, so everything is accounted for.
To answer the broader question of why we use the gravitational force for the gravitational potential energy, we need to define potential energy for an arbitrary conservative force. In general, the potential energy is the energy contained in the configuration of a system. What "configuration" means can differ for different forces; in the particular case of the gravitational force, this is the energy associated with the relative positions of everything in the system. Since, for our purposes, the Earth is fixed, the gravitational potential energy is the energy associated with the height of the object.
The other key piece of understanding is this: the work done by a given conservative force is equal to the negative change in potential energy of the object. (This comes from the more fundamental statement that force is the gradient of potential energy). Since gravity is a constant force $mg$ in the approximation you're using, the work done by gravity is simply the force multiplied by the distance the object moves in the direction of the force, which is the definition of height. From this you get that the work done by gravity is $-mgh$, so the associated change in gravitational potential energy is $mgh$. If we choose to set the gravitational potential energy of the object at $h=0$ to be $PE=0$, then we have our formula $PE=mgh$.