We fear our own deaths because we've forgotten our true nature, so we've attempted to banish the specter of death from society as a whole. I'm willing to bet people who have this reaction to the death of a chicken are being brought in contact with their own mortality, and that scares them. Given that we are all going to die, we should get comfortable with death rather than hide from it.
Hint: they don't run around in happy green fields.
Some of them do. I actually check the code on eggs before buying to make sure I'm getting those from free-roaming chickens.
Not to mention that farmed chickens are bred to be so fat they can barely walk anyway: https://www.wired.com/2008/02/chickens-cant-w/
Me, I figure there's all sorts of tasty critters in the world and I'm willing to try every one.
The use of the organic label is to distance people from this imagery by constantly repeating the propaganda that everything is being done 'nicely'. However, actually showing any kind of killing, no matter how nice, destroys that propaganda. The key trick is to use the correct words to make sure people do not conflate what they are eating with anything negative, and the organic label is part of that.
When you go deeper and attempt to explain how the killing is better, you simply open the audience to the fact that there is still killing being done. This is the wrong approach for the target market. It's better to distance the brand from the actual realities as much as possible.