The old Fed database has all shotguns coming in around 99% one shot stops, regardless of choke, load, brand or gauge.
As I've written before, even the inadequate for most hunting 410 has the ME of the old 41 Magnum Police load, an admirable stopper at short range.
I've Brenneked a few deer with some 20s, and the blood trials were no longer than with a 12 and that 73 caliber roto-rooter.
Ammo selection is more limited, accessories like mag extensions are a little harder to find, and the lighter 20 often simulates the kick of a heavy loaded 12. A six and a half lb shotgun putting out an oz of lead at 1225 FPS feels the same no matter what the gauge.
As for the 870 YE,it's not just lightness,but that short stock has a pad with a smaller surface area to spread out the push. Frankenstein, my parts 870 12 gauge, weighs within a few oz of the YE and SEEMS to kick much, much less. YMMV, kick is quite subjective.
Oh yes, with proper 7/8 oz loads, the little YE is less of a bronc and more of a pony.
HTH...