I think the saw tooth metal tube idea is great, should be easy to make esp if you had access to a Dremel type tool to cut the teeth and the broken screw would act to keep it all centred,,,the only problem might be how long is the old screw? Would be worth knowing this before you start, maybe you could drill one or two holes through the shaft using the finest of your drill bits to see how deep it is, I doubt this would have much effect in it stength esp as you could tap in a thin sliver of wood and glue afterwards?