Question: Does light really travel more slowly near a black hole?

It is a routine problem for beginners in general relativity to calculate the coordinate velocity of light for the Schwarzschild metric. Starting from the metric:

$$ ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \frac{dr^2}{1-\frac{r_s}{r}} + d\Omega^2 $$

We use the fact that light travels on a null geodesic so $ds^2 = 0$. This immediately gives us for a radial light ray:

$$ \frac{dr}{dt} = \pm c \left(1 - \frac{r_s}{r}\right) \tag{1} $$

\begin{align}
D_{\mu} A^{\nu} &= \partial_{\mu} A^{\nu} + \Gamma^{\nu}_{\rho \mu} A^{\rho} \\
\Gamma_{ {\color{red} \mu \nu}}^{\rho} &= \frac{1}{2}g^{\rho \lambda} (\partial_{ {\color{red} \mu}} g_{ {\color{red} \nu} \lambda} + \partial_{\color{red} \nu} g_{ {\color{red} \mu} \lambda} - \partial_{\lambda} g_{{\color{red} \mu \nu}}) \\
\Gamma_{ \lambda \, {\color{red} \mu \nu}} &= \frac{1}{2}(\partial_{ {\color{red} \mu}} g_{ {\color{red} \nu} \lambda} + \partial_{\color{red} \nu} g_{ {\color{red} \mu} \lambda} - \partial_{\lambda} g_{{\color{red} \mu \nu}}) \\