It's a very subtle point! The trick here is that you need to be careful when talking about the reference frames.
To an observer at the infinity, a clock at the core of the Earth will tick slower than a clock on the surface of the Earth because the "core clock" is sitting in a more curved space, and that's it.
The difference between the clock on the surface of the Earth and the clock at the core is that the surface clock can't follow the "straight lines" (geodesics) in that curved space. So it experiences acceleration due to the force of inertia. And the thing preventing that movement is the repulsive force between atoms that make up the bulk of the Earth.
If this repulsive force magically disappears, then the Earth's atoms will immediately start moving at the straight lines, in trajectories that will lead them all into a point at the center of the Earth.
To add: the force of inertia due to moving in curved lines instead of geodesics depends on the "steepness" of the curved space. Which decreases as you reach the center of the Earth. So you get essentially the same result as with the classic Newtonian gravity, but through an entirely different path.
To an observer at the infinity, a clock at the core of the Earth will tick slower than a clock on the surface of the Earth because the "core clock" is sitting in a more curved space, and that's it.
The difference between the clock on the surface of the Earth and the clock at the core is that the surface clock can't follow the "straight lines" (geodesics) in that curved space. So it experiences acceleration due to the force of inertia. And the thing preventing that movement is the repulsive force between atoms that make up the bulk of the Earth.
If this repulsive force magically disappears, then the Earth's atoms will immediately start moving at the straight lines, in trajectories that will lead them all into a point at the center of the Earth.
To add: the force of inertia due to moving in curved lines instead of geodesics depends on the "steepness" of the curved space. Which decreases as you reach the center of the Earth. So you get essentially the same result as with the classic Newtonian gravity, but through an entirely different path.