I am in Ontario and I want to run computer networking wire from the basement to the second floor. The most direct route is inside the box that carries the furnace exhaust to the chimney on the roof. The wire would be outside the metal furnace exhaust but inside the wooden box that contains the metal pipe.
Any chance of running it outside the wooden box? In my old house, there was a small gap down they one side that I was able to push a metal fish tape down all the way to the basement.
I don't know about the construction of your house. Mine is mostly double brick, so I could just drop cables between the two skins. For the new parts of my house, I tied the ethernet cables onto other cables and pulled then up through the wall cavities. This sounds a lot safer than what you are looking at.
Ethernet is rated at 100m length so the most direct route may not be necessary.