The only reason for requiring a pump run 24/7 is to add O2 to the water. Once that is understood the next questions would be whether additional O2 is needed and then, if needed, what's the cheapest way.
Is O2 needed?
Depends on the size of the pond, its shape, how clean it's kept, temperature, size and number of Koi. The common answer is to run a pump 24/7. Fish do better with good O2 levels and it's commonly thought pumps add O2. But fish can survive in less than ideal conditions. In a large pond, cool temps, fairly clean, a few small Koi fed very little, no pump would probably be needed for the Koi to do well.
Cheapest O2
If additional O2 is needed to support a large fish load an air pump is a much cheaper source than a water pump. Air pumps can be in the 5-10 watt range. So that would cost about 42 to 84 cents per month for electric running 24/7. (5x(24*30))/1000)*0.117 = 0.42. And you could just have the air pump on hand if the fish showed signs of needing more O2 rather than running 24/7 on the off chance it's needed.
Gas exchange happens at the surface and has nothing to do with movement or "forcing air into water". Air pumps add a lot because each bubble creates a lot of surface area between the water and air in the bubble. Smaller the bubble means more bubbles, means more surface area which is why manufacturers tout bubble size.
So if you look over a still pond you will see a lot of surface area all with a continuous exchange of gases. Those leaving and those entering. The science is a bit complex but is called
diffusion.
For water pumps it's all about where the water goes. Just circulating water would add very little O2. Lots of people think ponds lake stratify like lakes but ponds are far too shallow for that. Over a small water falls would add some, but way less than people think. A large waterfall adds more. A long stream adds more. A fountain adds some O2. Using a water pump to push water through a venturi setup adds a lot of O2 but still at a much higher (10x) cost than an air pump.
So if you looked at the amount of power it takes to create a waterfalls vs that same power applied to an air pump you could run like 15 pumps for the same energy cost. Basically your whole pond would be bubbles...not that this would be a good idea. Too much O2 can kill fish too.