First of all

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so lucky said:
If you can stand one more opinion: As a rule 4 year olds do not understand the concept of death, nor the ethical issues involved. He does understand that when he hurts one of the birds, you pay a lot of attention to him. Have you been busier than usual lately, to the point he may be feeling the need to do something to get you to talk to him, talk about him, think about him, react to him? It's not an actual planned course of action, but kids do what works for them, and he has learned pretty quickly what gets your attention.
I wouldn't put too much importance on what excuse he gives or what he says when you question him. He is just trying to think of what you want him to say, and probably doesn't know why he did what he did.
The good thing is, kids can change their behavior pretty quickly and easily. It may not take much at all to divert him.
But I think I would consider an electric fence or something to keep the kids from the birds. If you have one younger than 4, you still have several years before they are responsible enough to handle tiny birds safely. It's not that they are mean; they haven't done the brain development yet to know and do the right thing. By giving them responsibility and freedom too early, you may make it harder on them to mature appropriately.
If you and the kids can see a therapist for a few sessions, they may be able to help you come up with a plan.
From a developmental point of view, this is spot-on.
A four-year-old simply can't be left alone with creatures that helpless, and as an aside, shouldn't even be left alone with most dogs (there was a recent tragedy near us where a small child was killed by the family dog, a perfectly lovely dog that had never acted aggressively in the past).
Four year olds don't make much of a cause-and-effect connection to things around them, that's why you need to remind them over and over again about things. Death truly isn't a fleshed out concept at all at this age,
especially if he watches a lot of cartoons--think about it, the coyote falls off a cliff and then gets right back up again and goes on chasing the roadrunner. As adults we get it, it's supposed to be funny, but to kids it is just mind boggling if they have any concept of death at all.
Also, what everyone is telling you about receiving attention is spot-on as well--it's called negative reinforcement, getting attention for doing something "bad".
So then we get to the "what should you do" about it. The negative reinforcement needs to be replaced with positive reinforcement--you need to "catch" him being good/helpful etc. and reward that behaviour with your time/attention. You also can't let him be around the animals unsupervised. At all. Or it will happen again.
YMMV, and I absolutely hate giving parenting advice on topics this serious, but this is all based on my background in education and developmental psychology and all offered freely--
always remember that free advice is only worth what you paid for it, none of us is in your shoes or knows your child or the whole situation.