Hence 'c' for that photon is effectively "zero" in both proper frame at that location and at emote frame of remote observer
Can you map the surface of the Earth on a flat two-dimensional map without distortion? No.
Can you map a curved universe with a single instance of four coordinates, which you can think of as a map between a four-dimensional manifold $$\mathcal{M}$$ and a flat four-dimensional image in coordinate space, $$\mathbb{R}^{\tiny 4}$$, without distortion? No.
The Schwarzschild
geometry is static and spherically symmetric, and hard to work with because we can't put our hands on it, mark it, cut it up, etc.
The conventional Schwarzschild
coordinates look like spherical coordinates + t but they describe the world of coordinate space.
The Schwarzschild
metric describes the space-time of the Schwarzschild
geometry in terms of the Schwarzschild
coordinates so that we can use the tools of algebra and analysis in coordinate space and say something about the Schwarzschild space-time
geometry.
Using the Schwarzschild coordinates the coordinate speed of light can be practically read off the metric since for any beam of light ds = 0.
Thus we have $$\left(1 - \frac{r_s}{r} \right)^2 c^2 = \left( \frac{dr}{dt} \right)^2 + \left(1 - \frac{r_s}{r} \right) \left(r \frac{d \theta}{d t} \right)^2 + \left(1 - \frac{r_s}{r} \right) \left( r \sin \theta \frac{d \phi}{dt} \right)^2$$
So for radial light we see that the coordinate speed is a function of position $$\left(1 - \frac{r_s}{r} \right) c$$ and for orthogonal light (dr =0) we see that the coordinate speed is $$\sqrt{1 - \frac{r_s}{r} } c$$ where $$r_s = \frac{2 GM}{c^2}$$. This anisotropy is a consequence of the distortion caused by using a mapping to flat coordinate space. Another example of that is that the coordinate speed of light goes to zero at $$r = r_s$$, but we know by general covariance that the speed of light is physically
c at all points in space-time.
But just like the Mercator projection is not the only way to make a map of the so the Schwarzschild coordinates are not the only choice for the Schwarzschild space-time geometry. Generally, one is free to pick any point in the manifold and any standard of rest allowed by physics (i.e. time-like), and construct the Dirac normal coordinates where locally all geodesics look like straight lines. This roughly corresponds to an airport's polar map so that distances and angles near the airport are well-preserved.
Photons are not "accelerated towards bh horizon". They travel at c=constant.
Clearly this is nitpicking, since light curves under gravity and stationary observers closer to the black hole see falling light as blue shifted relative to observers far from the black hole. There is even a radius exterior to the Schwarzschild black hole where light could orbit in a circular path.
Acceleration means more in physics than just "goes faster."
