what TM said. You need to know how deep your ice typically freezes and make sure you have enough depth below that, if you want to successfully overwinter your fish (along with keeping a hole open). And if your pond has any depth ABOVE ground, you have to take this into consideration. Usually, the deeper ponds are easier to overwinter. Mine is 44" and ice rarely exceeds 15". I am one of those that went to a pond breather simply because it is easy and effective. In the beginning, I used an aerator for the hole (worked the first two years, no problem) until one winter the line got pinched and I lost its functionality.
I don't run my pumps/waterfall in the winter for fear of ice dams and diverted water. I don't want problems then as accessibility and working conditions are a pita. All that said, if you have few fish, have a good amount of surface area, good amount of water volume, have minimal detritus/decaying organic matter in your pond, and keep the ice shoveled to allow light to penetrate, you don't even need the hole (but I'd do it anyhow, just in case). My winters can be as low as -10 but typically stay in the teens during the coldest months.
Btw, you're not using the aerator to insure sufficient oxygen for your fish (cold/winter water holds more than warm/summer + your fishes' needs go way down in winter), you're doing it to allow gas buildup from below to escape through the opening. That's why if you have little decaying organic matter, you're ahead of the game.