Proud ground swatter here. I also usually wait for my deer and moose to stop moving before shooting them as well.
I use a boito backpacker, 410, 12 inch barrel while on the ATV, it is good out to 20 yards and stores inside my locking box when I get to my moose hunting area. I also use a 20 ga single baikail or 22/20 ga savage, as the 20 ga shells are cheaper than 410 and for when the grouse start getting skittish.
I always go for the head shot and find it a point of pride in the hunt camp to present a limit of grouse meat to the gang for dinner with no pellets to eat around.