At the center of the earth an object should experience weightlessness because gravitational force would exist equally in all directions. Does this mean that the depth in the earth's interior with the most pressure would be somewhere between the surface and the center of the earth?
No, because omnidirectional pull is not effectively the same as lack of pull. The lack of an imbalanced force does not cause any lack of force, per-se. Therefore, pressure remains even if the pressure gives the illusion of weightlessness due to being even.