Killing the champion isn't a problem. Some champions would actually really like that (Yorick, Aatrox).
The problem is - what has come before that?
You have champions with so many possible storylines, and so many possibilities in the impact they can leave on the world. If you kill them right away, there's no point in having this character.
I've always said I wouldn't mind Syndra dying. She's human, and she has a target on her back. But don't kill her off right after she got out of jail. If a champions has had their time, a good story, so long a they have done something in their life, sure. Death is not the problem, it's how much development came before that.
To get back on Syndra since she's my prime example - if she dies after doing nothing, killed by a stray arrow or a random assassin, then yes, it will feel bad, and it won't be poignant or anything. It'll just be that, bad.
Killing characters? Yes. I believe that's what make good stories. If you can't fear for your fave's life, there's no real stake.
Butchering characters? No. They deserve their time to shine. I'm not saying everyone will all reach their goal, as some champions have opposite goals. But let them have stories before you kill them. Let them get that sweet character development.