The easiest way seems to be to take the integration variable to be $x= |\frac{z_1+z_2}{2}|$ and then integrate over the position of the point furthest from the origin. This gives
$$ \operatorname{exp\_abs}(2) = \frac{\int_0^1 \mathrm{d}x \, x^2 \int_x^1 \mathrm{d}y \sqrt{1-y^2} }{\int_0^1 \mathrm{d}x \, x \int_x^1 \mathrm{d}y \sqrt{1-y^2} } = \frac{64}{45\pi},$$
where the answer was kindly provided by Mathematica.