I'd love to hear an explanation for this ^ statement, because it just doesn't sound reasonable. Oxygen is required constantly, and I would expect the lack of it to become apparent almost immediately. It isn't analogous to poor nutrition, which causes problems that gradually come to light over weeks or months. A lack of sufficient oxygen will affect an organism immediately, right now! Fish removed from a well-aerated, well-oxygenated aquarium and placed into a bucket for temporary confinement or perhaps transfer, can if overcrowded and/or overheated begin gasping at the surface within a very short time...certainly not "hours"....by the time fish are surfacing, oxygen has likely been low for hours.
I would suspect that part of the problem is that old bugaboo: overcrowding. Personally, I keep my fish in lightly stocked tanks and ponds, such that a power outage is not immediately followed by fish deaths due to lack of oxygen. Aquarists today always seem to need one more fish...and this results in crowded aquaria that are totally dependent upon artificial means of aeration, filtration and other forms of water management. Then, when these artificial methods are interrupted, disaster ensues.
Referring to ponds raises another question: what exactly is meant by a "pond"? People smugly talk about "growing out" fish in an aquarium and then moving them into a pond when they are large. Then they show a picture of the "pond"...and it's a puddle that can be stepped across easily, or a small kiddie wading pool better suited to bathing a dog. Calling a container holding X gallons of water by the grandiose term "pond" does not confer magical fish-keeping qualities upon it. It's still X gallons of water.
My single small outdoor pond is a cavity dug into the earth, waterproofed with an EPDM liner...and that's it. It has a light stocking of fish, a heavy growth of plants...submerged, floating and emersed...and a maximum depth of perhaps 36 inches in the center. I have toyed with a cheap solar-powered pump which did not last long, but that was strictly an experiment in aesthetics. The fish have never displayed any apparent signs of oxygen deprivation. In any case, the pump only functioned during sunlight hours, whereas all those evil oxygen-consuming plants engaged in their heinous activity during darkness, reverting to producing oxygen rather than consuming it before the pump ever got going in the morning...just as they do in nature...
Being an inground pond, mine has its temperature buffered and moderated by the surrounding earth. It will never warm up to the point where oxygen becomes a concern, at least not with the stocking it has. In fact, the calm water...which the worryworts among us would characterize as "stagnant"...is several degrees cooler at the bottom than it is at surface, and during hot weather the fish tend to stay in the depths. They make periodic feeding sortees to the surface but spend most of their time deeper...just as they do in nature...
Contrast that to an above ground "pond" like the several stocktanks in which I raise fish outdoors during the warm months. These tanks are also "stagnant", and while they undergo temperature stratification, they do not have the protection afforded by being dug into the ground. Surface temperatures can rise far higher than the inground pond ever experiences. Warm water extends deeper down, and in any case they are only around 22 inches in depth. Water temperatures in even an 8-foot round tank can vary as much as 12-15 Fahrenheit degrees in the course of 24 hours. These tanks are also heavily planted and lightly stocked, but they are much closer to the edge of disaster at any given time. They require shading from the afternoon sun during the height of summer...and I'm talking about summer in Manitoba, not California or Florida.
Fish behave in these tanks the same way they do in the dug pond; they spend time near the bottom during the hottest periods, approaching the surface in mornings and evenings...much like they would do in nature. No gasping, no distress.
Aeration added to these tanks must be approached very carefully. Simply tossing in an airstone or a sponge filter...as I originally planned to do...can lead to disaster. Destratifying the water with aeration produces a uniform temperature throughout, top to bottom, and that temperature can be far too high at midday in full sun. I have had problems with several species that simply could not do well at these temps, which can approach or exceed 80F easily and sometimes hit 90F. As non-intuitive as this may sound...the only solution is to discontinue aeration and circulation and allow the water to stratify again by temperature, allowing the fish to find their comfort level...much as they do in...well, you know.
Oxygen dissolves in water naturally; aeration will obviously increase the speed at which this can occur, but there is much calm water in nature that is full of healthy fish. How can that be?
If you love gadgets and tech then you can play with it all you want, and it can absolutely improve aspects of fishkeeping...but that doesn't mean it's necessary, nor even always desirable. If a reliance on tech support encourages you to create an unnatural overcrowded system that falls apart when the power goes down... that's not a good thing.