I think we often fall into the ALL OR NOTHING mentality with this. For example, "Your shotgun either puts all pellets into a coffee cup at 25 yards, or it is an alley broom.". That is a mistake in my opinion. Just like sights or no sights, moving or stationary, there are degrees.
A shotgun...for me is a weapon ostensibly intended to kill nefarious human adversaries intending to do likewise to me or mine. I don't play gun games. Nothing wrong with them, but its just not my thing.
Much of the shotgun lore today comes from the efforts of Jeff Cooper and Gunsite. Cooper was first and foremost a rifleman. And in my opinion, he looked at every small arm from that perspective. Even the entire Modern Technique curriculum is basically an effort to run the pistol like a rifle. When Cooper and his people began looking at the shotgun, they took the same perspective.
Shots at Cooper's shotgun class (which I attended with my duty SGN of the day) were focused on extending the impact of the shotgun as far as possible. Interesting to note that we did shoot clay birds in the rifle class, but not in the shotgun class.
Just as guns get modified to fit a shooting match, so did guns there get modified to do well in that class. On came the ghost rings for those 150 yard slug shots, the side saddles and buttcuffs for those shoot-one-load-one drills, and slug-select drills, and the choked barrels to keep the scores up there. Eventually, just as most others things gun gurus write about, it becomes almost a religion to his followers.
A tight choked Modern Technique shotgun will do very well at gun school shooting problems. But at the price of losing the close range capability of the weapon.
I worked in a very densely populated area in SoCal, and got into a bunch of gunfights. Many of those were with a shotgun. Let me tell you of some of what I saw.
Shotgun Gunfight #1 - Bad guy shooting people on a beach at 0300 HRS. Shot fired by me at 15 yards in low light (moonlight). Bad guy hit with 10 of 12 pellets. DRT.
Shotgun Gunfight #2 - Bad guy robbing a Store. Bad guy running and shooting at us - shot at 3 yards. Spread of impact bypassed his ballistic vest. DRT. Low light - parking lot lights.
Shotgun Gunfight #3 - Running gunfight in parking lot (actually LAPD's Parker Center's Parking lot ...but that is another story). Bad guy running away shooting Glock 21 at me. Two shots fired...one hit him with partial pattern, on the run, in extreme low light. Second shot dropped him. Distance was 10 yards.
Note - The natural spread of the pattern is what allowed the hits on those bad guys due to the low light, fluidly changing situation, the rapidly moving fight, and the incoming shots. The circumstances precluded anything but a "mount the shotgun and shoot him NOW!" method.
The shotguns used here were all off-the-rack Remington 870s with standard 4 shot magazines, wood stocks and cylinder bores. Ammo was Winchester OO Buck 12 pellet magnum (no reduced recoil loads in those days). So that is my frame of reference about what a shotgun needs. I think its a mistake to try to turn it into something it is not.
What will serve you best is a naturally expanding pattern that allows you to hit under circumstances where you would miss with either a single projectile weapon, or a shot pattern modified to act as a single projectile. Accept the shotgun for what it and let your tactics make it as applicable to a varied number of situations rather than attempting to turn it into a rifle.