Why doesn't a neutral solution at any temperature have pH 7?
As the temperature changes the value of $K_\mathrm{w}$ changes(as you wrote) and then the concentration of $\ce{H^+}$ and $\ce{OH^-}$ also changes and now the concentration of $\ce{H^+}$ will not remain $10^{-7}$ and hence the neutral point shifts. Therefore, at any temperature the $\ce{pH}$ of neutral point is not 7.