The Christoffel symbols are not tensors, which specifically means that even if there is some coice of coordinates which makes them vanish, there is some other choice of coordinates for which they don't.
In terms of equations of motion, this means that for a given choice of coordinates, even though the space is flat, geodesics might not be given as $r(t) = (at+b, ct+d, et+f, gt+h)$ for real numbers $a, b, c, d, e, f$, even though there is some coordinate system for which they are.
However, the Riemann curvature tensor, given by $$ R^\rho_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma - \Gamma^\lambda{}_{\mu\sigma}\Gamma^\rho{}_{\nu\lambda} $$$$ R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma - \Gamma^\lambda{}_{\mu\sigma}\Gamma^\rho{}_{\nu\lambda} $$ is a tensor, so if there is a choice of coordinates for which it vanishes, then it vanishes in all coordinate systems.