Death sentence strikes me as an idiotic punishment.
That sounds like malicious abuse to me.
> and does not understand that this could kill them, or at least thinks they will not.
It would still be manslaughter at least, if not murder. And if shaking the baby was itself felonious abuse, then an accidental death resulting from that is probably felony murder in states with such statutes.
It suppresses the rate of vigilante justice. If the courts can generally do a better job than the vigilantes themselves, then society comes out ahead.
If we were to live in a society where parents can abuse their children to death and receive nothing more than a scolding from the state to be more careful in the future, then we'd have a lot more people killing each other in retribution. And once that starts in earnest, it often forms unending blood feuds. At the very least, imprisonment for manslaughter is warranted. For the state to mete out no punishment at all would be an abdication of their duty to deliver justice so that others don't have to seek it themselves.
If this was manslaughter/murder of an unrelated adult, then I could see that there could conceivably be some friends or relations inspired to vigilantism, but this thread has mainly been about manslaughter of children by parents due to incompetence.