Why don’t fixed gear bikes have brakes?
The reason is historical.  Fixed gear bikes have been traditionally used for track racing, long after most other bikes had switched to using freewheels.  In the high-speed velodrome environment, it would be dangerous if the rider in front of you could brake suddenly:  you would crash into them, and likely most of the group behind you would join in the fun (this is also true in a paceline or peloton on normal roads).  So brakes have always been banned, and aren't really needed on the track anyway.

But track bikes sometimes get retired to road use.  And as fixed gear has gotten more popular, many frame makers are building similar bikes for the road.  Some of them are closer to track bikes (no brakes), others might include front brakes or both front and rear.