I'd want to compare linguistic intuitions here. Of a recently dead person I should be inclined to say : 'She is dead'. Of someone who died last year I'd most likely say : 'She died last year' or simply (again) 'She's dead'. After a disastrous explosion I could say : 'Ten people have died' but I don't think this means or implies (in context) 'Ten people are known to have completed the action of dying'. Dying is a process; and being dead is a state. The state follows instantaneously on completion of the process. There is the completion of a process but none of an action.
You add : 'it may be tricky to claim that something that doesn't exist really has some property, like having completed dying'. I agree that one can't attribute a property to something that doesn't exist but I would draw a distinction between truth-claim and property-attribution. What does not exist cannot have a property. None the less it can be true of X, who no longer exists, that X died. This is only to claim that a certain past event occurred, 'There was a person named 'X' to whom the event of death happened'. That doesn't attribute a property; it only states a truth.