Ah, so there's a problem with my picture. That's the acceleration of gravity, which means it's a graph of the curvature, not the depth of the gravity well. The well is deepest in the center.

So it's certainly observable that the deeper you go under ground, the more time dilation you would experience.

The problem is that the interior of the Earth is a complicated bunch of stuff that we barely understand. There are convection currents which move material out from the core and back into the core, at least for much of the interior (or so we believe). I should clarify that we know things like the Pacific plate is sliding to the West and being dragged down by its own weight into the Mariana Trench. The crust gets deposited back into the molten mantle, and volcanic activity pushes molten mantle back to form new crust. So we know there are convection currents near the surface of the Earth, albeit slow moving currents. We know less about what's going on in the deep interior near the core. We still don't have good models for what creates the Earth's magnetic field, but it is widely suspected that vast underground currents of molten Iron are the root. If these exist, then there is significant mixing of the molten material inside the Earth, and it is not a rigid body.

The problem with making an actual calculation is that it will depend on what factors you count and which ones you deem insignificant in order to come up with a number. So the notion that the core of the Earth is 2.5 years younger than the surface is dubious without knowing the error bars on that calculation and what factors were taken into account.

That said, it is entirely plausible that the interior of any massive body is some finite positive number of years younger than the surface of that body, provided the body is rigid.

So it's plausible that the interior of the Earth is younger than the surface. If there is a solid (rigid) core at the center, then it's more than plausible, it's a given.