Say you can feel your hip flexors are fatiguing, so you want to take the 9:00 to 1:00 part of the pedal stroke easy for a few revolutions. Your quads/glutes (the big, strong muscles) feel fine.
On a freewheel you have two options:
A) Stop pedaling all together, and coast. Your hip flexors do zero work, but until you turn them back on, none of your other (potentially fine-feeling) muscle groups do either. you slow down considerably.
B) Push lightly through that part of the stroke with your hip extensors (or compensate by pulling harder on the equally fatigable hamstrings), then back on the power with your quads. Your hip flexors use less energy than normal, but don't get to rest entirely.
On a fixed gear you can do option B, or option C) Turn off the hip flexors entirely and "coast" through that part of the circle, and still have the pedals get around to 1 o'clock for the power stroke.
The same is true for any other combination of fatigued or fresh muscle groups. To coast on a freewheel robs you of the opportunity to maintain speed by compensating with other muscle groups. To "coast" on a fixed gear lets you work which groups you want.
Again, it takes the exact same amount of work to ride up a hill regardless of DT. If you can pedal a freewheeled bike without ratcheting the freewheel then it makes no difference when pedaling if you are riding a FG or SS (assuming equal DT friction). You don't need a fixed gear to do anything you are describing. The exact opposite muscle group of the oppsosite limb (according to your figure) still rotates the pedals in the situation you have described. If the hip flexors relax, the extensors of your opposite leg (which "feel fine" accorind to your scenario) will continue to rotate the pedals...thats why one phase is called recovery phase and the other is called the power phase....you have two legs. Because you have two legs, you can easily pedal complete circles with no muscles contractions during the recovery phases, even with a freewheel (millions of people do this everyday). To argue that a fixed gear is necessary for this is absurd. If you know how to pedal in complete circles without ratcheting the freewheel (everyone can do this), then having a fixed gear serves no advantage when climbing.
The perceived advantages of climbing with a FG are largely illusory. In fact it is a disadvantage if you allow the pedals to move your legs through any part of the pedal stroke, because not only are you loosing the same momentum that you would by coasting, but you loose additional momentum by having the drive train drain continue to drag your legs in circles. Loosing momentum means that you are going to have to expend more energy in the next pedal cycles to regain the momentum. If you loose more momentum each cycle, that means you also need to recover more momentum each cycle during the power stroke or you will eventually come to a stop. Its far more efficient to maintain as close to a constant velocity as possible.