The gravitational force would indeed be zero and a pendulum clock would not work. But the gravitational redshift between to locations is given by the potential

The gravitational force would indeed be zero and a pendulum clock would not work. But the gravitational redshift between to locations is given by the potential difference, not by the local force. The potential difference between the surface and center of the earth can be calculated using Gauss' law and comes out to be 1/2gR, where g is the gravitational acceleration at the surface and R is the earth radius. As a result, a clock at the center of the earth would run slower than one at the surface by 3.5E-10.