I doubt there is a real difference in mechanical advantage between a fixed cog and a freewheel. Back in the early 20th century, there was a feeling that the convolutions a chain undergoes in a derailleur system introduce an unacceptable degree of friction, but that complaint is no longer taken seriously.

The fundamental issue is that human legs produce maximum power over a fairly narrow RPM range, and having multiple gears allows the rider to stay within that range over a variety of riding conditions. That said, fixed gear training in the off season has long been popular, and can broaden the RPM range over which your legs can produce maximum power. Not to mention, riding fixed gear is fun and satisfying on its own for many people.