The real question is, "what exactly is a divergence?" The physical meaning of the divergence of a point particle's geodesic doesn't make sense. A divergence is meaningful for a vector field, for example the velocity field of an incompressible fluid. A velocity field moving radial outward from the origin does indeed have a non-zero divergence. Let's take a closer look.
A vanishing divergence for $\hat{x}$ motion
In Minkowski space in cartesian coordinates all of the Christoffel symbols are zero. For a velocity vector aligned with the $x$-axis, $\vec{u}\rightarrow(u^t, u^x, 0, 0)$:
$$ \nabla_\alpha u^{\alpha} = \partial_\alpha u^\alpha + {\Gamma^\alpha}_{\alpha\beta}u^\beta = \partial_\alpha u^\alpha.$$
This relation holds even if $\vec{u}$ is not a geodesic of the spacetime. In the case that $\vec{u}$ is a Minkowski geodesic (constant velocity), this vanishes.
If we interpret $\vec{u}$ as a uniform velocity field for a fluid, this makes sense. All of the fluid flows in the same direction, not spreading out. If we pick an arbitrary cartesian direction for the velocity $\vec{u}\rightarrow(u^t, u^x, u^y, u^z)$, nothing really changes. The fluid still all flows in the same direction, just some diagonal direction.
A non-zero divergence for $\hat{r}$ motion
In Schwarzschild space the Christoffel symbols are non-zero, but if you take the limit $M\rightarrow0$ some of them will vanish leaving the Christoffel symbols for Minkowski space in spherical coordinates.
Let's calculate a divergence assuming a radial velocity, $\vec{u}\rightarrow(u^t, u^r, 0, 0)$:
$$\begin{align}
\nabla_\alpha u^{\alpha} &= \partial_\alpha u^\alpha + {\Gamma^\alpha}_{\alpha\beta}u^\beta \\
&= \partial_\alpha u^\alpha +
{\Gamma^t}_{tr}u^r
+{\Gamma^r}_{rr}u^r
+{\Gamma^\theta}_{\theta r}u^r
+{\Gamma^\phi}_{\phi r}u^r \\
&= \partial_\alpha u^\alpha + \frac{2}{r}u^r.
\end{align}$$
There are four non-zero Christoffel symbols of the Schwarzshild metric that appear in the sum. You can look them up and see that two vanish in the limit of $M\rightarrow 0$. As pointed out in a comment, the ${\Gamma^\theta}_{\theta r}$ and ${\Gamma^\phi}_{\phi r}$ do not vanish, leaving an extra term.
This is the usual result for the divergence in spherical coordinates. The $r$-part of the divergence is:
$$\frac{1}{r^2}\partial_r(r^2 u^r) = \frac{r^2}{r^2} \partial_r u^r + \frac{2r}{r^2} u^r = \partial_r u^r + \frac{2}{r}u^r.$$
If we had kept the $u^\theta$ and $u^\phi$ parts of the of the velocity vector there would be additional non-vanishing Christoffel's giving the rest of the usual divergence equation (modulo the non-normalized basis vectors in Schwarzschild coordinates compared to the spherical coordinates used in E&M).
So what does an $\hat{r}$ aligned velocity mean? At all points in space the velocity field points radially outward. When a fluid moves radially outward, each streamline points a different direction. This velocity field is physically different than the $\hat{x}$ aligned one from before. Far from the origin $r\rightarrow\infty$, neighboring streamlines barely spread out at all. At the origin the divergence blows up, telling us there must be a source of the flow there.
In order to make a parallel, divergenceless flow in spherical coordinates we need to transform the $\hat{x}$ aligned flow to spherical coordinates for all points in space. This would result in many places having local velocity vectors that have non-zero $u^\theta$ and $u^\phi$.
