What am I doing wrong in finding the phase angle of power?
The mistake is in the definition of apparent power, which should be:
$$ \overline{S} \triangleq \overline{V}\overline{I}^*$$
In this way, by substituting for the current you get
$$ \overline{S}= \overline{V}\frac{\overline{V}^*}{\overline{Z}^*}=\frac{|V|^2}{\overline{Z}^*}$$
so that the phase of the complex power is simply the phase component of the impedance.