Step 1: match a constant with a constant.
The equilibrium constant $\mathrm{K_c}$ of a cell reaction is a fixed number for a given reaction at a given temperature, it does not drift around. So it can only be linked to another quantity that is equally fixed.
Step 2: see why E°cell qualifies.
The standard cell potential $\mathrm{E^\circ_{cell}}$ is defined at fixed standard conditions, so it too is a constant. The thermodynamic links $\mathrm{\Delta G^\circ = -nFE^\circ_{cell}}$ and $\mathrm{\Delta G^\circ = -RT\ln K_c}$ both use these constants, and joining them gives a clean fixed relation between $\mathrm{E^\circ_{cell}}$ and $\mathrm{K_c}$.
Step 3: see why Ecell does not qualify.
The actual cell potential $\mathrm{E_{cell}}$ is not constant at all. By the Nernst equation it depends on the real concentrations of the ions, and as the cell works these concentrations keep changing, so $\mathrm{E_{cell}}$ keeps falling. A value that is always changing cannot be tied to a single fixed $\mathrm{K_c}$.
Final answer: $\mathrm{K_c}$ is a constant, so it relates only to the constant $\mathrm{E^\circ_{cell}}$; $\mathrm{E_{cell}}$ varies with concentration during discharge, so it cannot be tied to $\mathrm{K_c}$.