Birds are almost certainly going to be one of the key survival foods for you after the SHTF. And here is why. All birds are edible worldwide. They don't all exactly taste great, but most do. Birds that eat fish typically taste nasty, but most birds eat seeds or insects so that shouldn't be a problem.
Another reason that birds will be part of your diet is because of availability. After all of the deer and small game has been hunted to such low numbers that they are no longer going to produce often enough to sustain you, you will have two options. Move... or switch food sources. If you have devoted a lot of sweat equity into building a new home you might not be too keen on moving. And this is where birds shine.
It will be IMPOSSIBLE for you to wipe out all of the birds in your area. Birds are very mobile as a group. Sure, you can knock down the local population of the less mobile/migratory of the birds, but there will be periodic influx of migratory birds such as ducks, black birds, robins, etc. that new birds are regularly showing up. Surely the bird population will take a hit as people fight for survival, but the vast majority of the bird species will not go extinct. As more and more people die to starvation, mob murder and disease the bird population will rebound.
As with all animals I suggest trapping instead of hunting. Trapping is just more efficient and frees up your time for other duties. Here is my link to bird traps and how to make them.
Traps
Here is information on how to clean / dress/ prepare your bird properly.
Dressing Birds