to me, since the light is always traveling at speed c, it is not too important to ask "at what point" does it have maximum speed
it has the same speed, namely about 300,000 km/s, all along its trajectory.
But it is only at minus and plus infinity that the velocity is purely radial.
at other points it has a (possibly small) transverse component
but none of this matters because your main question is what is the
maximum radial velocity, and I believe the answer is c, or approx
This would also be the case for a small object like a rocket ship which is launched towards the massive object at some speed very close to c,
like 0.999 c,
the maximum radial velocity (which could be achieve arbitrarily closely) is again c.
In this case, of a material object, I am not telling you where on the trajectory it would reach top radial speed, but simply that the least upper bound is c.