One of the few reasons for a rear brake on a fixed gear bike (at least in my opinion) is for long descents. Modulating between front and rear can keep rim temps down and prevent blowout.

I've had one incident where I did a 300' decent in about .4 miles and had a tube fail when the rim got way, way too hot.

Other than extremes like that, a functional front brake and a bit of modulation from your legs will keep you safe. Learn how to stop with your legs tho, never know when a cable will break