Step 1: $K_c$ and $E^\circ_{cell}$ are both fixed at constant temperature.
The standard Gibbs energy links both quantities: $\Delta G^\circ = -nFE^\circ_{cell} = -RT\ln K_c$, giving $E^\circ_{cell} = \dfrac{0.0591}{n}\log K_c$ at 298 K. Both are constants for a given reaction at given temperature.
Step 2: $E_{cell}$ is a variable, not a constant.
Unlike $E^\circ_{cell}$, the actual cell potential $E_{cell}$ changes with ion concentrations according to the Nernst equation: $E_{cell} = E^\circ_{cell} - \dfrac{0.0591}{n}\log Q$, where $Q$ varies as concentrations change during the reaction.
Step 3: Only constants can be related to $K_c$.
Since $K_c$ is a fixed value at given $T$, it can only be equated to the equally fixed $E^\circ_{cell}$. The variable $E_{cell}$ reflects the instantaneous composition ($Q$), not the equilibrium position ($K_c$), so it cannot be directly related to $K_c$.