Every affinely reparametrized geodesic remains unit speed

In the global Cartesian coordinate on the Euclidean line, [F1] gives ${\mathrm{\Gamma }}^1{}_{11}=0$. The coordinate function of $\gamma$ is $x(s)=s$, so $\ddot{x}=0$ and [F1] makes $\gamma$ a geodesic. Its velocity is ${\mathrm{\partial }}_x$, whose norm for $d{x}^2$ is $1$, so $\gamma$ has unit speed.

The affine map $\mathrm{\ell }(t)=2t$ has nonzero constant slope and hence is a genuine affine reparametrization. By [F2], $\tilde{\gamma }=\gamma \circ \mathrm{\ell }$ is still a geodesic, but $\mathrm{\mid }\dot{\tilde{\gamma }}(t)\mathrm{\mid }=\mathrm{\mid }2\mathrm{\mid }\mathrm{\mid }\dot{\gamma }(2t)\mathrm{\mid }=2$, so it is not unit speed.

Thus the displayed curve and reparametrization refute the universal claim. The exact failure is the missing restriction $\mathrm{\mid }a\mathrm{\mid }=1$: slopes $1$ and $-1$ preserve unit speed, every other nonzero absolute slope changes it, and slope $0$ would give a constant geodesic rather than a reparametrization diffeomorphism. The witness is one-dimensional, has no finite parameter endpoints or degenerate interval, and is explicit, so no choice principle is used.