When someone says a 12 g recoils more, they obviously aren’t comparing similar weight and velocity loads. Gone are the days only one or two velocities and weights are available. Physics is physics - if anything the lighter smaller rifle will kick more, unless you’re just picking random shells off the shelf at Walmart.
Do smaller gauges pattern better with same weight of shot? No.
My buddy has a streak of aristocrat in him leftover from his big deal first marriage who wouldn’t be caught dead with anything less than a $15k Brittany sired by Sir Douche Dove Licker, and a 28 ga costing at least that much. I laugh when he drags out 1 oz loads doing 1400 fps and tries to say his gun recoils less, but it makes him happy and reminds him of the days when his ex father in law would talk down to him while on the family’s fancy Texas hunting ranch with the ex wife treating him like dirt whenever possible. Lol