You are absolutely right that manually updating hours is a huge chore and not surprisingly it is universally despised. Making estimates beforehand in scrum is also problematic because it doesn't involve the time spent in previous phases as you pointed out. Kanban, however, solves the problems inherent in scrum, and virtually all digital kanban solutions automatically track the time for you in each phase of your workflow.

So, it is possible to measure productivity relatively painlessly in kanban as it limits work in progress (discourages multitasking, reveals bottleneceks in your workflow). Kanban also takes idle time into account (ex: time spent waiting approval from the management or another dept). See http://flow.io/how-kanban-can-help-you-measure-productivity.... for a brief overview of kanban's benefits.

Disclaimer: I am the author of flow.io, a lean project management app based on kanban.