Just like golf, you can play with one club but better off having choices to fit the requirements. Now that should give you the incentive to buy more than one. Right ??
However, working with your criteria, I'd have to agree with the 20Ga. in 3". You can even take ducks with one although you would have to get lucky on geese. You will never outgrow a 20Ga. and neither will your kids.
Be Safe !!!