It's a fight but it can be done. I had an '87 Trek 560 frame with the same absurd cable routing. You do know that only the inner wire goes through the chainstay, not the housing, right? I know that seems like a silly question but I've heard of some owners trying to get the housing in there too

Be sure the end of the wire is absolutely smooth and free of loose strands and just keep wiggling it around while pushing up and back from the front. It wiil pop out eventually.

If you get too frustrated, you could fit a clamp-on cable guide to the chainstay and route the housing and cable under the chainstay like civilized frame do.