In my younger years it was not uncommon for me to load and shoot half a case or more a week on a single stage Pacific press. I had gotten a Western Field 12ga shotgun for Christmas when I turned 12 and it only took pop about a month to get me that press verses buying shells.
During this time I studied up on patterning and went through several loads, wads, and such until I had things where I wanted them. Being a Western Field, there wasn't much I could do about adjusting the gun itself, but the loads, WOW that was some interesting stuff.
A year or two back a good friend of mine got himself a nice high dollar 12 gauge to get into duck hunting. I told him now what you need to do is go out and pick up several brands of shells, in different configurations and pattern them to see what does the best. He gave me that deer in the head light look as if I had just fallen off the turnip wagon. Well just to prove my point he went along with me and I set up a paper at 40yds and had him fire away with it. He was dismayed at how this high dollar shotgun would hardly hit the paper. It took a LOT of explaining to get him to realize that not only was it the load, but the fit of the gun as well that was screwing him up. Since it was one of the nicer guns, it had spacers and such which not only allowed adjustment of the LOP but also the angle of the fit of the buttpad. We played with it and got it down to hitting within the circle but he was still unsatisfied with the results. Finally after calling the manufacturer he was made aware that the European shotguns were made to shoot with the bead directly under the target, after which he started to blow the center out of the circles every time.
Like was mentioned, most folks around these parts simple buy a nice looking shotgun and head ot the woods. Most are never aware of the intricate adjustments which can be made to greatly impove the way they handle as well as shoot. Just like an expansive target rifle however, even the cheapest shotgun can do a LOT better when properly tuned to the loads and shooter.
As for the circle and dot thing, if you don't want to see the circle simply use a pencil to draw it, and you will not see it from 40yds. Use the stick on for centers. Just get a string, and tie a loop in both ends that measure out to 30" before you head to the range. Use a nail to hold it centered and the pencil to draw your circles.