A body is resting on the Earth's surface, where $h=0$. It is not likely to possess kinetic energy, because it is in rest. As potential energy is $mgh$, and here $h=0$, $mgh=0 J$. But a body cannot have no energy at all. What kind energy does this object possess?
You define $h = 0$ to be at the earth's surface. That's an arbitrary definition. What if we define it to be slightly below the earth's surface? Now it has a non-zero height, and therefore it has potential energy. In fact, we could even define it to be slightly above the earth's surface, and then it would have negative potential energy!
Converting between gravitational potential energy and kinetic energy is simply a convenient way of converting between height and velocity in a gravitational field. When you do the calculations, you'll find that only the changes in energy (i.e. work) are important. How much gravitational potential energy it has does not matter in the end, so it doesn't matter where you define $h$ to be.
So, what I am saying is that your statement that it "cannot have no energy" is actually incorrect. It has neither kinetic nor gravitational potential energy, but that's just a side effect of where you defined $h = 0$ to be.
Note: There are other types of energy it probably does have, like thermal energy (unless it's at absolute zero), but I don't think that's the main takeaway here in this context.
The gravitational potential energy is stored by the system comprising the Earth and the body and depends on the separation between them.
If the separation between the Earth and the body $h$ does not change much compared with the radius of the Earth $R$ then the change in gravitational potential energy when the separation changes by $\Delta h$ is $mg\Delta h$.
So when there is no change in height there is no change in gravitational potential energy.
If you want to call $mgh$ the gravitational potential energy of the body then height $h$ is the height above a position where you have arbitrarily chosen a height where the gravitational potential energy is zero.
So the change in gravitational potential energy is $mgh-mg0=mgh$ where $h$ is actually a change in height from a position where you have defined the height to be zero.