How can anything decay in such cold water and create a build up of unwanted gases if closed off? Why do we have to worry about keeping it open? What is the science to this? And when lakes are completely frozen over why doesn't that kill fish?
as Lisa noted, the ratio of fish load to volume/surface area is a lot different between lakes and our garden ponds. In fact, there ARE fish kills when the ice+snow exists for prolonged periods of time. I have a 2 acre natural pond at the end of my road and three years ago when we had a long winter and the temps in Feb reached record lows that did not relent until March, had one of these. The fish floated toward the shore and most were large. Between the lack of oxygenation and the toxic gas buildup, some can't survive. If you google fish kills in Michigan, you'll find reports of it happening even in larger lakes. But it takes the right conditions even then. IMO we depend on the typical January thaw to open up the waterways to allow this gas-off and re-oxygenation.
In our garden ponds, we're just guarding against the odds because our fish load to volume/surface area is a lot higher than the natural lakes. It doesn't take much to keep the hole open, so why not? And the best defense against winter is a deep pond with ample surface area. The smaller the pond, the more likely issues will arise. I have one client of mine with a 300 gallon preformed, probably doesn't go deeper than 18" and his gf seem to survive every winter. Gf are tough, more so than koi which are also pretty tough.
And as Lisa also noted, the decay process continues as there are bacteria which can survive colder temps. The activity is less but still on-going. This process uses up oxygen too, that's why having a healthy biofilm made up of algae is important because it too will keep putting out oxygen as photosynthesis continues. Of course this needs light to actually penetrate the ice, hence why fish farmers tend to recommend removing some of the snow on top the ice. I think once the ice reaches 5", the light is diminished too much, so it helps to keep the snow off as well. That said, I haven't done this yet and my fish seem to make it alright. I have lost one or two in various years and I put it down to a weaker fish/variety. I started using an aerator to keep the hole open then switched to a pond breather as it is a more secure system, imo. Easy too.
I suspect Lisa is having success with her system because the underground water tank is probably a lot warmer than her pond water would be if without. The earth's temp increases as you go deeper and if I remember correctly, tends to stay about 50 F down below the frost line. This means any pond that is dug in will get the benefit of this heat. Lisa's tank is getting that. I'm not a big proponent of keeping the water moving as I don't see any benefit, though her plant situation likes it better. For me, the risk of the pump and or system getting a leak or ice dam/bridge and emptying the pond is too high for little benefit. I do think the water is warmer at the bottom, even if only by a few degrees and I don't like mixing the top and bottom. I have a suspicion that if Lisa took the temp of her water, she'd find it is above what one would expect and that surely helps. Again, pond/lake fish are adapted to surviving this and as long as we guard against catastrophe due to our overloading of our ponds, the hobby has a great return!