How about going downhill, wouldn't you think coasting would save energy more energy than having to pedal constantly? Also, expand upon the whole going up hills on a fixed gear is easier. A freewheel bike can have momentum to and you are pretty much "connected" (as you put it) to a freewheel drivetrain up hills if you keep pedaling.
Of course being able to coast down hill would save a lot more energy with the rider just sitting on their butt the entire time but if you've ridden fixed gear/ track long enough, you learn to spin your legs where you are able to completely free up your legs spiritually and let them spin effortlessly to the same effect.
Going up hills fixed is easier because you are able to use your weight (yes, as with any bike) to push a foot/ pedal down whilst bringing up the opposite BUT thanks to the inability to coast/ being directly interconnected to the drivetrain, your other/ opposite foot has to go up no matter what, setting you up perfectly to use your weight again without having to pull up with the other. With a freewheel you don't get this type of interconnectivity/ momentum from the wheels.