Just because we weren't shown the transition with the reapers doesn't mean they weren't there. Reapers don't help a soul move on. They are the sole reason that a soul moves on. That's what we've been told and shown since Season 1. Remove the reapers, like in "In My Time of Dying" or "Death Takes a Holiday", and then people aren't able to die, even if they're suffering from a mortal wound.