Sorry, you will never prove that swimming faster burns less air. You may be able to prove efficiency can increase speed without burning more air.
Well, I actually was implying your second sentence above. Through better hydrodynamics, you can be more efficient, thus allowing for faster swimming speeds while at the same time using air at a lower rate.
But, it is also possible to be more efficient even without hydrodynamic improvements, lending a potential falsehood to your first statement as well. To prove this, just look at the limits. Take a diver at rest. What is his air consumption per foot traveled? It is infinite. Now take the same diver swimming at 60 fpm. In that case, the cubic feet of air used per foot traveled will be a real number, i.e. - non infinite. Faster can be more efficient!
You also will never convince me what I want to SEE can be seen better at a faster pace. To even assume that you could know what I want to see takes a level of arrogance (or ignorance) I don't even want to consider. I will give you the benefit of doubt that you are not doing that and you are just meaning more efficient diving will be better, which I can at least agree to in theory.
Hydrodynamic efficiency simply opens more options; more options for things to see, to follow, to get to, places to be or to extract yourself from. It is well within the scope of operations to go as slow as you want, including hovering at a standstill or crouching in the sand. However, as soon as you start moving, regardless of how fast or slow that is, a diver with poor hydrodynamics will always have to produce more physical power to move at any given speed than would a diver with better hydrodynamics. This is true at 30 fpm, and also at 120 fpm. At some point the dirty diver will limit out and have no more power available to go faster.
What do you do if you find yourself in a statistically common current moving at about 200 fpm? Give up, and hope someone else saves you? That's what divers generally do these days, and most of the time, if they are diving from a boat with a good crew it works. But, it doesn't always work out. With good hydrodynamics, I can swim at 240 fpm and still make 40 fpm progress against the current. That may be enough to get me to where I need to be to extract myself from the situation without incident. If it doesn't work, I still have the option to quit and hope someone picks me up, but I would consider that situation a failed dive plan, not a standard M.O.