I've seen that Vaidya metric does not use the Schwarzschild metric, but instead to get it you simply go to Eddington-Finkelstein coordinates and there put mass as a function of retarded time. From
$ds^2=\left(1-\dfrac{2M}{r}\right)du^2+2dudr-r^2\left(d\theta^2+\sin^2{\theta}\right)$
We can go to
$ds^2=\left(1-\dfrac{2M\left(u\right)}{r}\right)du^2+2dudr-r^2\left(d\theta^2+\sin^2{\theta}\right)$
Is it because the black hole is not the same after time translations, and therefore the $g_{0i}$ components of the metric would not be zero in Schwarzschild coordinates or is there another motivation?