A rear sway bar will reduce roll and reduce tire clearance issues. And the rear bar doesn't have to be "HUGE" to do it either.
Before I added a rear sway bar on my Duster I would frequently get some rubbing on the inner wheel tubs when I cross axled through a dip or over a speed bump - the body roll was causing the interference. I added a 7/8" Hellwig sway bar and no more rubbing in those situations. Obviously if I manage to get the body to roll that much again the tires will rub, but the rear bar eliminated all the rubbing during normal driving conditions.
You don't want to go too stiff with the rear bar, as you'll end up in oversteer mode. I prefer a little bit of oversteer, but you can definitely have too much of a good thing. And you want your car matched fairly well from back to front. In my case, my rear leaf springs are quite a bit softer than the front suspension, so the rear bar was needed to match things up and balance the car too.
And a rear sway bar will not stop the axle from shifting side to side under the car. Depending on how it's mounted it might add a small amount of resistance, but we're talking a very small amount, just what the bushings are giving you when you're pulling the end links sideways. Definitely not the reason you'd want to add one.