The effect is real, not an artifact of our instruments. And it’s not really correct to say time slows down more and more as you get closer to the earth’s core. Instead it slows down as you move to regions where the gravitational field is greatest or strongest. At the core itself, the strength of the field is zero: you’d float at the core if you hadn’t been burned up and crushed by the conditions down there, and your clock would run as fast as it can anywhere in the vicinity of the earth. I’ve seen some really great quotes about this that makes this clear. I’ll try to find some of them and put them up here for you.
Since we can’t actually reach the core, I was using the core more as a reference point to distinguish between the high peaks & low depressions on the earth’s surface, and that the affects of gravitational pull was the real factor. Since the surface of the earth is the point of gravitational pull, then yes the gravitational pull would decrease as you continued towards the center mass of the earth and when you got to true center the gravitational pull left, right, up & down would counter each other to result in a net zero gravitational pull. As you are being puled in all directions at the same gravitation pull.