Do you need a12 ga?....
It all depends on what you plan to do. And how well you want to do it. And what compromises you are willing to accept to do it.
The bottom line is that ALL shotguns operate in the same approximate velocity range. There is an upper limit to shot speed, that when passed, patterns literally get "shot to hell".
To keep the ability of the gun to put shot on the target in a decent pattern, the only way to increase the "power" is to throw more shot. Which also has the benefit of a larger pattern.
So, it would seem that the "best" shotgun is the largest one you can comfortably manage. And that means 12ga, for most shooters.
BUT, there are many times when the "best" isn't any more useful than "good enough". And there are situations where things other than how much shot can be thrown determine what is best, as well.
Ammo choice, and cost. Size and weight of the gun itself. Recoil.
20ga is really fine for a whole lot of things, 20ga guns are normally built on frames that are smaller and lighter than 12ga guns. This means, in some people's hands there is a noticeable difference in ease and rapidity of handling.
3" 20ga shells can throw the same weight of shot as the standard 2 3/4" 12 ga. (of course, same mass, same speed, lighter gun means a difference in recoil).
Ammo availability, local to you. Regions where the 20ga is popular will have stores stocked better. Smaller places might only carry one or two different 20ga loads vs 4 or 7 (whatever) 12ga loadings if the 20ga isn't locally popular.
Lots of things to consider, these are only some. Some will advise, "get both" and point out the places where one serves better than the other. If you can afford it, the perfect solution to the dilemma, right?
The only issue with that (other than cost) is that if you own both a 12ga and a 20ga you must constantly be on guard against the "12/20 Burst".
Its not much of a concern if one is diligent about paying attention to which ammo is used, but could be a concern where the gun(s) might be used by others. If someone drops a 20ga shell into a 12ga gun, it will slide down the barrel and get stuck a ways ahead of the chamber. Put a 12ga shell in that chamber and bad things will happen!
This is why, traditionally, 20ga shells are bright colors (yellow, orange, etc) where 12ga shells are red, blue, green, black, etc. So you can see the difference easily. (not every ammo maker everywhere follows this rule, but most do).
I keep only two sizes of shotgun in my house, 12ga, and .410. For me, these do everything I ever wanted a shotgun to do. I have nothing against the 20ga, I just don't have any personal use for one.