Question

why is the equilibrium constant related to E°cell and not to Ecell?

Answer 1

Nernst equation actually relates the emf of cell with the reaction quotient.
$aA+bB\longrightarrow cC+dD\\ Q=\dfrac{[C]^c[D]^d}{[A]^a[B]^b}\\ E_{cell}=E_{cell}^{\circ} -\dfrac{RT}{zF}\ln{Q}$
The value of $Q$ may or may not be equal to $K$, so the reaction may or may not be in equilibrium at the given instant. When the reaction attains equilibrium, the feasibility of both forward and backward reaction is zero.
$E_{cell}=0,Q=K$
Putting these values in Nernst equation we get,
$E_{cell}^{\circ}=\dfrac{RT}{zF}\ln{K}$