Ponds in nature don't have pumps. I also have seen goldfish living in golf course ponds, that were large and healthy and had no filter, pump, or any aeration device installed. Yet, pet stores sell this equipment.
Why it necessary to have any of this equipment, when in nature fish seem to do well without that?
In nature the fish only live in these ponds that have a good quality. The other ponds have either only plants and algae, or are more muddy, stinky places without higher forms of life.
If you have a pond in your garden, there are a lot of things to have in mind.
I will give one example: if the pond is small and it gets warm in summer time, the water gets warm too. Only big amounts of water stay cool from morning until evening. Then, if the water is warm, it could not store enough oxygen to let the fish breathe. That could make an aeration useful. Another thing caused by too warm waters are red tide/algae bloom. Then the algae will use all oxygen (at night all plants use oxygen instead of producing it) and the fish could not breathe anymore.
Another point is the pee and poo of the fish and other nutrition, for example flushed in by the rain. If you like to have a nice amount of fish, it would be more than a likewise pond has in nature. In nature a little amount of fish would make poo and pee and the plants in the pond use it as nutrition. But if you like to have this in you own pond, you will have a lot of more plants than fish and you would probably never see the fish. Additionally, you need to exactly know the amount of food for the fish. Because if there is more food than the fish could eat (or they eat it and make more poo and pee), the plants have more nutrition and grow and grow and the algae could grow too. Then in the night, when the plants use oxygen too, it could be that there is nothing for the fish left.
In conclusion, you have a lot more of work if you like to have a "natural" pond. Failures are not acceptable at this setting. And because of that a lot of people use auxillary equipment, like pump and aeration systems. Even in nature, there are times when a pond or even a lake gets red tide and then has to develop itself new. But this is not something you'd like to have in your garden :)
A pond usually has many more fish than a natural pond. Likely much fewer plants in a home pond than a natural. And artificial food is added. And, as answered, in warm weather the higher fish population are likely to need more oxygen, especially Koi. In a natural pond there is water movement caused by wind and possible stream input/output. The aeration and filters try to replace the oxygenation of natural water movement. Although, in addition to my conventional pond, I do have about a 300 gallon plastic tub with a few mollies (I think) that has no other equipment; and I do not add food. It does have tadpoles. That is, with a few of the right fish and very sparse feeding, you do not need the equipment.
