Of course air bubbles do provide active sub-surface aeration. But there is more to the process of aeration than a simple general statement. The actual efficiency of sub-surface aeration whether by air stone or diffuser is determined by the depth of the body of water and the size of the bubbles. Even the most efficient system will not add much Oxygen by transference from the rising bubbles in comparison to the Oxygen transfer achieved at the water-atmosphere interface as a result of the mixing action imparted upon the water column by this same column of rising bubbles. So essentially Brokensword is correct in his assumption.
Waterfalls are actually more efficient at adding Oxygen to a body of water but the downside is that this oxygenation is confined to an area immediately surrounding the waterfall and is not spread evenly throughout the water column. This is one of the reasons that one will find fish congregated at the base of a waterfall. It is not that the pond has low Oxygen but that the water in this area is much more heavily oxygenated.
A few assorted cites-
"However, almost all of the oxygen dissolved into the water from an air bubble occurs when the bubble is being formed. Only a negligible amount occurs during the bubbles transit to
the surface of the water."
Water Aeration---Wikipedia
"Many People are surprised to learn that the majorioty of oxxygenation occurs through the water's contact with the atmosphere, relatively little increase occurs through direct
diffusion from bubbles."
Lake Notes--Illinois Environmental Protection Agency
"Synergistic airlifts are at the other end of the spectrum because they expend very little energy and circulate huge quantities of water upward. In this case, the interaction at the
surface of the lake adds to the majority of the oxygen added to the water."
Pentair Aquatic Ecosystems
"Fine-bubble, bottom-laid aeration also moves the most water at the lowest cost. By producing a gentle, laminar flow of small bubbles, at a consistent rate of rise, water
molecules are moved easily to the surface, where oxygenation occurs."
and-
"Aerators that create coarse bubbles, such as air stones and flex diffusers, employ an uncontrolled release of air, which has been shown to be much less effective at moving water compared to fine-bubbles.
Principles of Aeration- EPAeration