Assuming the eigenvalue of position operator $\hat x$ equal to $k$, can I not write:

$$\begin{align}
\langle\psi_n|x|\psi_m\rangle &= \langle x\psi_n|\psi_m\rangle \\
&=\langle k\psi_n|\psi_m\rangle \\
&=k\langle\psi_n|\psi_m\rangle \\
&=k\delta_{nm} 
\end{align}$$

But I know that $\langle x \rangle =0$ in case of even potentials (I don't know how that happens) and what I have written above is wrong, at least in case of even potentials.

Taking the example of infinite 1D square well, the states are :
$$ \psi_{n} \left(x\right)=A\sin\left(\frac{nx\pi}{L}\right)dx $$
then,
$$ <x>=A\ int_{-\infty}^{\infty}sin\left(\frac{mx\pi}{L}\right)x\sin\left(\frac{nx\pi}{L}\right)dx $$
If m=n=1,
$$ <x>=A\ int_{-\infty}^{\infty}x\sin^{2}\left(\frac{x\pi}{L}\right)dx $$
if we apply an even potential then the equation gets reduced to$$ <x>=\frac{1}{L}\ int_{-L}^{L}x\sin^{2}\left(\frac{x\pi}{L}\right)dx=0 $$ while in case of a potential(neither even nor odd), the equation leads to $$<x>=\frac{2}{L}\ int_{0}^{L}x\sin^{2}\left(\frac{x\pi}{L}\right)dx=L/2 $$ ?
Here n=m=1 but $$ <x>=0 $$ for even potentials,which is confusing me! It should be $$ k\delta_{nn} $$ right?