$$\require{cancel}$$
$P = \frac{1}{c} \int_{4\pi} I \cos^2 \theta \ d\omega , \label{4.6.1}$
If the radiation is isotropic, this is not zero; it is $$4\pi/(3c)$$. In the expressions for $$J$$ and for $$P$$, the power of $$\cos \theta$$ is even (0 and 2 respectively) and one can see both physically and mathematically that neither of them is zero for isotropic radiation. One the other hand, the expression for $$F$$ has an odd power of $$\cos \theta$$, and it is therefore zero for isotropic radiation, as expected.