there's no --- ZERO --- mechanical benefit with fixed vs. free wheel. Also keep in mind that a multi gear derailleur system cannot be a fixed gear.

When you ride a freewheel, the unit has no moving parts as you pedal, so for all practical purposes is the same as a fixed wheel. The ONLY difference is that a freewheel allows you to coast if you wish to, while a fixed doesn't.

That said, old timers and coaches do find advantages to fixed gears. It's not mechanical, and is more about helping riders improve cadence and smoothness of their pedaling style, and breaking the habit of coasting, which many do unconsciously. Over the years, I've dealt with riders who couldn't break the habit of coasting over bumps or around curves. I'd put them on a fixed gear bike, and within a short time they'd be cured and could even pedal through hairpin turns without touching then inside pedal.