= Solution
In geodesic coordinates built from an orthonormal basis, $g_{ij}(0)=\delta_{ij}$. Every radial curve has coordinates $x(t)=ta$, and substitution into the <geodesic equation> gives
$$
a^ia^j\Gamma^k_{ij}(ta)=0.
$$
Taking $t=1$ proves $x^ix^j\Gamma^k_{ij}(x)=0$.
Conversely, assume the metric and Christoffel-symbol conditions. For every $a$ in the star domain, $x(t)=ta$ satisfies the geodesic equation and has initial velocity $a$ in an orthonormal coordinate frame. Uniqueness of solutions to ordinary differential equations gives $\phi^{-1}(ta)=\exp_p(ta)$ wherever defined. The star-domain assumption covers all of $U$, so $\phi$ is precisely a geodesic coordinate chart. This is the <Radial Christoffel-symbol criterion for geodesic coordinates>.
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