I don't know about being drowned in a Have-a-Heart trap but we're keeping chickens here not giving coons advanced degrees in chicken slaughter. Yes, new coons will eventually replace the coons that you remove. That is why coons are a renewable resource and predator management and control is environmentally friendly. Perhaps a better question is not "I don't think I would like to be drowned in a cage, would you? but "If I were a chicken I don't believe that I would like to be pulled up against the wire on my coop late at night and be literally eaten alive.
Like Dori says in
Finding Dori: "There is always another way."
About the "pulled up against the wire and eaten alive" bit, you can easily prevent that by using hardwire cloth that racoon paws can't reach through and a solid surface, like the corrugated metal I used, on the lower part of your run. See photo.
I guess I have a real soft spot for elephants - don't ask me why, I don't know - but I caught myself realizing that it is sort of insane of me to expect African farmers to leave room for the native species, like the elephants, and for me to not even have the ingenuity / motivation / initiative / requirement for myself to leave our American native species some room to live, even as I encroach on their habitat.
I don't want my birds killed, but that does not mean I have to kill racoons. There are other options that work in most cases.
Agree with centrarchid that a picture of the particular property or area to be protected may help to see what ideas may work in your case to live and let live and protect your birds.
If I were eating racoons, I would agree that I need to kill them, but if I just want to keep them away from my birds, I don't see why I need to kill them and I would prefer not to. The words 'Do onto others as you would have them do onto you.' are often on my mind and I would rather be sparked and walk away or warned off by a dog I can see and avoid than to be trapped and drowned or shot because I walked into a wrong situation just trying to get some food for myself.
The dogs, by the way, have not killed any racoons. They keep them away by their presence, scent marking, barking and chasing them off when they get too close to the outside of the fence. So far so good.