I favour a lightweight 12 gauge - and there are more than a few available. In my case it's an upland styled over and under with 26" bbls and interchangeable chokes. When I get too old and feeble to handle the weight of a shotgun like that, I guess I'll have to resign myself to purchasing a golf cart to drag my pathetic ass around and learn how to play golf.
Everything else being equal you will get better patterns out of the bigger gauges than the smaller ones due to the shorter shot string. I think it is important to pattern your shotgun to learn how it patterns at various distances and find the loads that pattern the best - no matter what game you're hunting. Furthermore, there is no law that requires you to shoot heavy shot loads in 12 gauge shotguns. If you aren't a reloader, there are some manufacturers out there selling 5/8 oz. 12 gauge loads, using 71/2 shot; should be just fine for quail, huns, and bunnies.
If you're a shotshell reloader, there's loads as light as 3/4 oz. at 1100 fps out there, and then all the way up. If you aren't a shotshell reloader, you're missing out - I started out with a Lee Load-All. Not the fastest thing in the world, but they cost somewheres around $40, so will pay for themselves pretty fast if that's a concern. And best of all, you get to decide what size shot, what weight of shot, how fast, etc. You can tinker the loads until you find the best possible patterns in your shotgun.
On the flip side of that, there is also no law that says you can't stuff some heavy loads in that light 12 gauge should the hunting scenario make that suddenly desireable. But you can only load up a 20 gauge, 28 gauge, .410, etc so much...
My rabbit hunting isn't done in terrain that is any uglier than where I hunt quail and pheasants - in fact, it's the same place. So I don't need a beater shotgun that will take the abuse. Your situation, of course, may differ.