angular momentum - Energy in the physical pendulum

I am considered the problem of the physical pendulum, in particular the derivation of the energy of the physical pendulum. I am slightly confused by one point.

Looking at the derivation given in this link,

it seems to be that they only considered the gravitational potential energy and the rotational kinetic energy. However the bod is not rotating about its centre of mass, therefore should we not also account for the translational kinatic energy of the centre of mass?

If we did this, then the final expression for energy would have an additional factor of $\frac{1}{2}Mv^2=\frac{1}{2}Ml^2\omega^2=\frac{1}{2}I\omega^2$; so then the final expression for energy would be

$mg\frac{d}{2}(1-cos\theta) +I\omega^2$? Where am i going wrong here?