Track bikes must be brake-less. If one racer suddenly decelerates, you have a pile-up.
Brake-less on the street is a pointless and unsafe affectation.
There are literally no hills where I ride. So I have only a front brake (yes, the front is the most important). When I lived elsewhere, I had front and back.
Brakes. I rarely rode fixies, but when I did, I tended to build up more speed than my legs could take away. And with some of the hills here in King county, that is bad!