Why have a brake on a fixie if you can have a brake on a freewheel single speed? A TRACK BIKE WITH A BRAKE AND FREEWHEEL SINGLE SPEED WITH A BRAKE LOOK EXACTLY THE SAME.
Why have a brake on a fixie?
Fixies can do things that a bike with a freewheel can not. Mostly these things are about "tricks" (riding backwards, etc.) and forcing you to pedal rather than letting you coast (which I'm told has training advantages.)
A brake on a fixie, especially a front brake, can stop it better than backpressure on the pedals, and it's far less rough on your tires, so it's certainly an advantage there.
If by "freewheel single speed with a brake" you mean a coaster brake, then they
don't look exactly the same. In particular, the coaster brake almost always has a reaction arm.
If a cop hassles you, you can probably just say you have a coaster brake -- pedal back to stop -- and unless he knows bikes he won't know the difference, unless he asks you to demonstrate. But if he does know bikes, the lack of reaction arm will be obvious, and he'll have noticed the lack of it already before he starts hassling you about not having brakes.
To make a long story short, I ride a brakeless track bike for the reliability, theft resistance
Reliability? Is it really more reliable than the same bike with a front rim brake added?
As for theft resistance, does anybody really believe that thieves can't ride a brakeless fixie? They'll be more likely to crash, sure, but then again ... so are you.
If you want a bike that the inexperienced have a hard time riding, try a tall bike, or a SWB USS recumbent. Or a swing bike.
also just because it was different from my 2 other free wheeled bikes.
Now, that reason I have no arguments with.