Most of the stuff people say about riding fixed gear bikes just isn't true:

It makes you stronger/teaches you to spin: you pedal as sloppily as you otherwise would, except you can't coast, instead of learning to put power through all parts of the crank rotation. That might point out to you where your spin is dead and help you work on it, but it's the you working on it, not a fixed wheel that does it.

It makes you "more aware": like the above, this is only true if you were already unaware in the first place, and then it's a lazy aware, reacting to constant sudden death instead of riding in a way where other people's horrible mistakes and miscalculations are the only ones you have to worry about

You have better "traction": in slippery conditions, you have worse traction and speed control, assuming you ever ride in any conditions where you will hit spots where any pedal force will spin the wheel, whereas coasting allows you to "float" through a bit. Fixed gears don't help you here, but hurt you, but momentum and bike control ability do.

You can control your speed better than with a brake: same as all the above again I guess, except moreso, unless you just go really slow all the time, but I'll skip the ad infinitum brake arguments here

Singlespeeds have moving parts: Most of my bikes have been cheapo junk over the years, and riding year round in lousy conditions in a city with lousy roads, the only parts I've had suddenly fail were because of poor maintenance or hasty installation. That's a big your-mileage-may-vary, but I rode a way used (bought from someone who used it for ten years) lx freehub for years in snow and rain, and when it froze up, I took it apart to find the cones were majorly scored and the freehub needed to be oiled. Put it back together with lots of grease, and someone else is using it now. I've had more catastrophic fork failures or rear brakes locking because of snow and ice than a greased bearing-aided ratchet failing by about 5:1.

I do believe that fixed wheels help you in climbing unsteep grades that aren't very long (say half a kilometer or less?) in that you lose less momentum with less input required. I consider myself a strongish climber for an untrained/uncompetitive biker, and this was the first thing I really felt. Also, trackstands are easier on a fixed gear bike, but it's like hopping curbs or bunnyhopping - if you can do it, you can do it, it. I learned on a geared bike and can do it as long as I can on a fixed bike, just with more effort and rocking, but trackstanding is only useful or functional on a track, so that doesn't really count as an argument.

I also believe the fixed gear bikes are inherently much more dangerous when piloted by unskilled bike riders (as a large percentage I've seen have been), as they require more knowledge and ability and attention to ride as safely.

I ride fixed gear bikes (most of the time anyway) because it's fun. You can do weird stuff, like as I was coming up to my house, I figured it was slippery enough to climb off and stand on the left pedal and just skid through the snow to my door on one foot for like 20 feet. That's not functional, especially when you're doing it beside a picket fence. Or you can do the same when you're coming to a stop and just leave one foot clipped in and go up and down like a see saw for a revolution of the pedals. Come to think of it, that's pure function, if you're looking to look like a dork, which maybe is why I do it.