I think there is some truth to your theory, but not enough to really matter. There are other disadvantages that will offset the slightly tighter pattern which I discuss below.
Remington makes the 870 with a 21" barrel that takes choke tubes intended for turkey hunting. You can buy dozens of factory and aftermarket choke tubes to get any pattern you want. If I were to choose only one it would be Improved cylinder. It still shoots slug and buckshot well and gives patterns satisfactory for moderate ranges. The gun is light, short and makes for a good do-all gun. It isn't ideal for long range waterfowl or dove shots. But OK for most other shooting including self-defense. I have an older one in 12 ga and one of the new guns in 20 ga.
There is a lot of overlap in performance. You can get 20 ga shells with 7/8 oz of shot up to 1 1/4 oz. Either 7/8 or 1 oz are most common. You can buy 12 ga shells with 7/8 of ounce up to 2 ounces with 1 oz or 1 1/8 oz being the most common. You can make a 20 do almost the same thing by carefully choosing your ammo. Just be aware, to get close to 12 ga performance will require you to buy less common, and more expensive 20 ga loads. And if you need more than 1 1/4 oz of shot to get the job done you need a 12.
As long as you don't need more than 1 1/4 oz of shot to get the job done 20 ga will do almost the same thing but there are some disadvantages. You will ALWAYS get better patterns with a 12 ga even with the same weight shot. You don't just deal with the width of the shot pattern, but the length. 20 ga loads have a longer shot string. Meaning some pellets are going to be farther ahead of the ones in the back of the pattern. A 12 ga shell has a shorter shot string, meaning all of the pellets arrive on target at closer to the same time. If shooting a stationary target it doesn't matter. But when shooting moving game or clay targets it can matter.
Theoretically 20 has less recoil. But that only applies if you're shooting less shot in the same weight gun. With most guns a 20 is about 1 lb lighter than a comparable 12. If you're shooting 1 oz loads from a 6 lb 20 ga and 1 oz loads from a 7 lb 12, the 20 is going to kick more. If you're shooting 7/8 oz 20 ga loads in a 6 lb gun and 1 1/8 oz loads from a 7 lb 12, recoil will be about the same. The 12 only kicks more with heavier loads.
I switched to a lighter to carry, short barreled 20 ga for turkey hunting years ago. Since I'm shooting at a stationary target the long shot string isn't a handicap. With modern shells and aftermarket choke tubes patterns are good enough for 40 yard kills. A 12, would extend that range by about 10 yards. But to me the lighter gun to carry is a better choice.
For SD, or waterfowl hunting the bigger payload of 12 ga shells and the shorter shot string make a 12 my choice. I choose a 20 for the times where carrying a lighter gun is more important. If you don't waterfowl hunt then a 20 probably comes close enough to meet your needs. But I'd not want to be handicapped with a fixed cylinder bore.
