I start out with the mindset that nothing is immortal. One day, bar something happening to me, I am going to find the dead body of every single animal I now own. It's just a matter of whether it happens now or later. Being sad about it serves no function.
I raise some birds for meat, so I guess I have a tougher skin than a bunch of people. I was pretty wound up the first few times I had to butcher, but once I was over my fear of doing it wrong I could do it calmly. It doesn't really affect me now, though I get looked at like I'm crazy if I say that so I generally keep that bit to myself and play "sad".

I'd like to say that I am not flippant about their lives in any way. It is my responsibility to care for them and give them the best lives possible. Cruelty to an animal is despicable. When my flock management practices put birds on the chop list... I do the deed gently, with awareness as to what I'm doing, and use their bodies as food for myself so they have not gone to waste. My empathy just seems to come more from a moral code rather than fully from wherever everyone else is getting it.
I did, however, cry when I had to put down Chrissy, my favourite duck, a few years back. She was injured and I really thought she was going to pull through so it came as a bit of a shock. She got worse after another accident that broke her leg in a way I tried and failed to set, and well, I'm not cruel enough to have let her die slowly and in pain. I still felt like a dirty traitor because I'd promised her I would pull her through... sending her to get a bullet put in her head was tough. Kept thinking that maybe she'd get better if I waited a little longer, though I knew logically it wasn't true, she'd been going downhill for well over a month. Sudden losses don't have tough decisions to make, but they're not fun either. I usually dispose of dead birds by tossing them in the woods somewhere. They're usually cleaned up in a week as long as I put them on a certain side of the property. I've buried a few too, though that's only feasible in summer and even then the rocks and roots make it hard work.
Some people just plain can't compartmentalize the same way others can. Others can learn to.

Hard to give advice to someone else on that front when it's such a personal variable. Hopefully you come up with something that works for you.
The kids here aren't really attached to the birds so I have no advice there. I lost plenty of my own pets (not chickens, but rabbits, etc) back when I was their age, but I was an odd child, so advice that would have worked for me would likely not apply to them.
Also, two chicks dying in so short a time sounds iffy. Hopefully they don't continue to do so.
