Nah, you fire the coach because he hasn't won a playoff series in 5 years and we've only got a few more whacks at the apple. Kapanen is a symptom. Dumoulin riding the top pairing is a symptom. Carter being used in high pressure defensive situations is a symptom.
These are all correctable things, but we're not correcting them, and we're not winning. So, I don't get why you want to die on that hill, tbh. Like if we were playing out of our minds, then fair enough, Sullivan can run this team however he wants. But we're playing boring hockey, the top 6 looks like shit, and the blueline is atrocious. We can't replace all of that at once, but a new coach could maybe spark something and maybe squeeze a bit more blood out of this stone. I see it as a worthwhile gamble.