I avoid shooting 3" mag in any gauge, might OK in .410. Makers put 3" chambers in 20 gauge guns, and the 20 gauge seems to be sold as being able to shoot up to 1-1/4 oz, like a 16 or a 12. 'Often' to me means those loads are in the gun when hunting. That is what will be fired when a bird goes up. I know 3" loads are not needed for most hunting.
Thinking that 6 - 6.25 lb is what people like in 20 gauge gun, comfortable at 7/8 oz and 96x at about 1 oz. 6.5 I think is heavier than people usually want in a 20, but it would help with recoil esp. if heavier loads are used for hunting.
Rule of 96 calls for 7.5 lb for 1-1/4 oz, but if the gun is carried for a couple of shots at birds, stretching the rule might be OK.
1-1/4 oz at 1185 fps (20 gauge) seems easier to take than at 1330 fps (12 gauge). The 12 gauge is 12% faster than the 20, so maybe the 12 gauge gun needs to weigh 12% more than the 20. If 7.5 lb is tolerable for the 12 at 1330 fps (32 ft-lb recoil), then about 6.5 lb would be about the same for the 20 at 1185 fps (31 ft-lb recoil).
I think I could take, but not like, that recoil for a few shots spread over a day's hunting. 31 ft-lb is stiff.
A Browning Gold would be my choice if I intended to shoot 3" all the time. 6-3/4 lb (less recoil energy), gas auto (spread the kick over more time), level comb (does not attack the teeth) no pitch (does not drive comb up into face).