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Of course, working with affine parameters is of great advantage. That's why it's better to use the "square form" of the Lagrangian for the geodesics, i.e.,Dale said:
$$L=-\frac{1}{2} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu},$$
where the dot again means the derivative wrt. an arbitrary parameter ##\lambda##, but thanks to Noether's theorem, since the Lagrangian is quadratic in the ##\dot{x}## and since it doesn't explicitly depend on ##\lambda## the "Hamilton-like" conserved quantity ##H=L## is conserved along the solutions of the equations of motion (which are just the geodesic equation). This means that for the solution ##\lambda## is automatically an affine parameter along the trajectories of the particle.
Another advantage is that this works without trouble for both light-like as well as time-like geodesics. In the latter case you simply choose the conserved quantity ##g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=c^2##. Then you have ##\lambda=\tau##, with ##\tau## the proper time along the geodesic. For the light-like case you have to set ##g_{\mu \nu} =g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=0##, and of course there's no proper time, but ##\lambda## is still some arbitrary affine parameter. The physics of course doesn't depend on the choice of this parameter.