Something to consider
And that is the cross section of the shot charge in flight. Because, not all the pellets will be in a postion to hit a flying target at the same time.
A magnum 20ga may throw the same numer of pellets as a standard 12ga, but it has a longer shot string. What this means is that on a crossing bird (for instance) when the shot string gets to the bird, the 12 ga will have more pellets "on plane" with the bird, while the 20ga will have more "ahead" or "behind" the moving bird.
If you are on target, that doesn't mean much, but if you are a little off, then the 12 might mean a downed bird, over the 20.
But, this must be balanced against the lighter weight and handling of 20ga guns. Which could mean the difference between you being "on target" over the bigger, heavier 12ga. A lighter gun is less tiring to carry all day, and that might make the difference in your swing. OR, it might not.
A 20ga gun might make you a better shot than a 12, or it might not. Everybody is different that way. .410s are often used for beginners, because the light weight guns and light recoil make them easier to shoot. BUT, the small shot pattern makes them harder to hit sucessfully. They are better considered an expert's gun, at lest in terms of how easy they are to down birds. Again, if you are "on", you are on. But the area of being "on" gets smaller the smaller the bore size of the gun.
I'm not a big shotgunner (was nearly 40yrs ago, but not much since) and I only have two calibers, 12ga, and .410. They do everything I need done these days. Nothing wrong with a 20ga, if you use it well.
There is one thing you do need to watch out for, if you own both a 20 and a 12. A 20ga shell will slide down a 12ga barrel, and stop a couple inches ahead of the chamber. VERY BAD NEWS if you drop in a 12ga and fire it! Thats the reason most 20ga shells are yellow or orange, while 12s are red, green, black, or blue, so you can see the difference in low light.