Dogs have instincts, but that's no excuse for a dog to take it upon itself to do anything, especially killing things that it sees and knows live on your property with your permission. This all stems from a very shaky foundation, and the only way you're ever going to have a dog that you can trust around your birds is if you start repairing that foundation of respect. I say this over and over again. This doesn't happen from tying dead birds to necks or shock collars or hitting the dog. It starts with you commanding that your dog respects your direction. Most dogs in families do not do this and it seems to me that people complain about this but aren't really bothered enough by it to really do anything about it... or change their way of thinking and approach to their dog or dogs. If this offends you, I'm sorry... not my intention.
A trainer can show you how to teach a dog something, but they can't instill this foundation. Only you can do it for yourself and your dog.
Once you've established this foundation which, most all dogs are perfectly intelligent enough and capable, willing to adhere to, then you can work on necessary training around the birds teaching your dog that it can be around the birds but they are not food or toys... again, perfectly attainable regardless of instincts. Dogs are smart and have a lot of self control, believe it or not. And as I always say, if a dog doesn't actually have the ability to do these things, don't let it have puppies.
Too many dogs are being bred without this basic intelligence these days.. there was a time when dogs were only allowed to breed if they worked with people in a way that was helpful. The crappy dogs were dispatched. Anyway...
But then again, most people seem perfectly fine with a dog that runs their lives, whether they want to admit it or not. And that's fine... as long as these dogs aren't going onto other people's properties and destroying someone else's birds or stuff or biting their kids or killing their cats or whatever, as I always say. I have friends and family with dogs like this and there was even a time when I had a dog like this, but my life back then was hellish because of it and visiting these people is hellish. I don't bother with it anymore. And not just because of their dogs either.