As to the 20 vs. the 12, it's not JUST a gun weight issue.
The 20 has a smaller "footprint", allowing it to be carried "easier" even if it happened to weigh the same as a particular 12 ga. For those familiar with carrying a crossbow, it's like the difference in carrying a crossbow with very wide limbs vs. one with very narrow limbs. Rotational torque issue.
Next, the 20 ga shells are about half the weight of 12 ga shells even when using 1 1/2 oz loads in the 20. You shave off another 1/2 lb or so in your pockets just by carrying 20 rather than 12 ga extra shells.
And if you weigh guns fully loaded, you will find the weight difference loaded is more significant than unloaded, i.e. approximately another 1/4 lb shaved off with the 20 vs the 12.
Last but not least, I've found the 20 ga, despite being a lighter gun, actually has less recoil with 1 1/2 oz loads than my 12's typically have with 1 1/2 oz loads. This may not make "sense", so I welcome others' thoughts on this.
Not knocking the 12, which will always be ballistically superior to a 20 when both are using TSS.
But a 20 using TSS ballistically outperforms a 12 using copper-plated lead loads.