Yes, that is sad. I'm sorry your little chick didn't make it. The only consolation I can offer is that the chick was with you during its short life, and that you didn't let it suffer. If it had been in the wild, it would have died slowly and painfully.
We've had to put down grown chickens, and it isn't a pleasant thing to do emotionally. The whole time we are sharpening the hatchet, we are praying that we are swift, accurate, and strong with our "chop" so the
chicken will go quickly and as painlessly as possible.
If an animal seems too sick or injured to recover, we decide to euthanise it. It's just easier than letting the animal suffer.
I don't know if you are religious, but Proverbs says that the righteous man regards the life of his beast. I interpret that to mean not only its welfare during its life, but also knowing when it's more compassionate to just let it die as humanely as we can make it.