Not "most" bird hunting, as that would include chukar, pheasant, grouse, and waterfowl as well.
Dove and quail? Yes.
To get back to the OP's question, if you want a lighter gun, a 20 gauge is about as small as you can reliably go (and you still give up SOME performance). After that, you have to start using expensive, performance-enhancing ammunition like TSS to even attempt to replicate performance of the larger bores.
Now of course, there are plenty of people who will argue that a 410 or whatever works just fine for them. The fact of the matter is, you can see effectiveness drop off as shotgun bore gets smaller and smaller. Anyone who tells you otherwise hasn't watched enough birds fall or is lying. A 12 gauge is noticeably more effective than a 20 gauge on pheasants and waterfowl. A 20 gauge is noticeably more effective than a 28 gauge on chukar and grouse.
You can still have great days of shooting with the smaller bores, but the guys using those guns are picking their shots. All else being equal, moving to larger gauges enables you to make longer shots. Extra reach will put more birds in the bag, and more quickly too.