I tis very important to notice that you can do this at a given point, but not at any given curve. In fact, it is true that for geodesics you can define a locally flat coordinate system such that the Christoffel symbols vanish along the curve. This is compatible with the idea of the equivalence principle for $inertial$ observers.
However, for more general trajectories, it is not possible to define a such a coordinate system: the Christoffel symbols get corrections due to the acceleration of the trajectory, and these are compatible with the notion of fictious forces that are seen due to non-inertiality.
The way to analyze the problem above is by means of the Fermi normal coordinates (FNC). These are a coordinate system that is associated with a timelike trajectory and describes "what an observer undergoing this trajectory sees". In case you want learn more about this, I recommend https://arxiv.org/abs/1102.0529. In any case, it is possible to show that the metric in FNC can be written locally around the curve (up to first order in acceleration and curvature) as
$$\begin{aligned}
&g_{\tau \tau}=-\left(1+2 a_{\mathrm{i}}(\tau) x^{i}+R_{0 \mathrm{i} 0 \mathrm{j}}(\tau) x^{i} x^{j}\right), \\
&g_{\tau i}=-\frac{2}{3} R_{0 \mathrm{jik}}(\tau) x^{j} x^{k} \\
&g_{i j}=\delta_{\mathrm{ij}}-\frac{1}{3} R_{\mathrm{ikjl}}(\tau) x^{k} x^{l}
\end{aligned},$$
where $\tau$ is the proper time of the trajectory (and the timelike coordinate of the FNC) and $x^i$ denote the proper distance from the trajectory, which are the spacelike trajectories of the FNC. $a(\tau)$ is the trajectories proper acceleration and $R(\tau)$ is the curvature tensor along its motion.
The Christoffel symbols to first order in acceleration and curvature then read
$$\begin{aligned}
\Gamma_{i j}^{\tau} &=\frac{1}{3}\left(R_{0 \mathrm{ijm}}+R_{0 \mathrm{jim}}\right) x^{m} \\
\Gamma_{\tau i}^{\tau} &=a_{\mathrm{i}}+R_{0 \mathrm{i} 0 \mathrm{~m}} x^{m} \\
\Gamma_{\tau \tau}^{\tau} &=0 \\
\Gamma_{j k}^{i} &=\frac{1}{3}\left(R_{\mathrm{j} \mathrm{km}}^{\mathrm{i}}+R_{\mathrm{k} \mathrm{jm}}^{\mathrm{i}}\right) x^{m} \\
\Gamma_{\tau j}^{i} &=R_{0 \mathrm{mj}}{\mathrm{j}} x^{m} \\
\Gamma_{\tau \tau}^{i} &=a^{\mathrm{i}}+R_{0}^{\mathrm{i}}{ }_{0 \mathrm{~m}} x^{m}
\end{aligned},$$
which can be seen to vanish if $a^i = 0$ (inertial motion) and $x^i = 0$ (on top of the curve).
