I don't have any issue with controlling coyote populations. However, I do share some of the OP's concerns about how some people choose to go about it.
I've had to kill racoons, foxes, feral cats, and weasels (the coyotes don't like to come in close enough to the house to be an issue) that have been after my chickens/ducks, and I'd kill them all again. However, I've never gotten any pleasure out of it, nor have I taken it personally that they try to catch an easy meal. After all, if chickens weren't delicious and relatively easy to kill I'd probably raise something else
Every animal deserves as quick and clean a death as possible, and some hunters do seem to either like - or in more cases not care about - inflicting pain. I'm not going to pretend I've never screwed up a shot, but if causing another living thing pain doesn't bother you on some level there is something very wrong with you.
If you wound an animal - even a coyote - and fail to recover it, you should feel bad about that. And, if you're doing it on TV in front of a national audience you should probably be aware that acting overly callous about it is going to turn people - some of whom may even vote - off hunting all together.