Here is my reasoning for why I use a 3 inch circle.
That would not be ideal, or even sensible to use for any kind of fast moving shooting where you would be sweeping the target. Like grouse or pheasant or duck hunting.
We have very few grouse and pheasants anymore since the state in their infinite wisdom reintroduced coyotes to our area. They really screwed the grouse population entirely. To put it into perspective, I have seen a grand total of TWO grouse in the last 15 years. Further, I kill every coyote I have a chance to draw down on.
So, unfortunately, I don't get to hunt grouse anymore and now am excited just to see one because it's such a rare treat now that takes me back to my childhood hunting them with dogs.
But I do hunt turkeys and squirrels, which require tighter patterns, so I judge how well my gun is patterning by a small 3 inch circle.
The more pellets I get in the 3 inch circle at a given distance (usually around 20 yards or so), the happier I am.
Now if I was hunting flying birds, I would probably opt for a lot looser choke, like a modified or something along those lines so the pattern would spread out more at a distance, allowing more of a chance that I'll get it.
But the whole point of patterning your gun and even patterning it with different brand shells or shot size or choke, is to see where and how the pellets are hitting the target and to optimize the results for what and how I am hunting.