That video is as good a guide as any I've seen. It should have accomplished your task. I have had a tightish bend in the cable housing make it really hard to push the cable end through - I bent the solid cable end a little bit the first time I did that, by mistake, but still bent. I was able to straighten it and still use the new cable assembly.
You did mention in your post the best way to push the core through - once you have the cable housing installed through the transom and between the Y pipe and the flywheel housing, you just lay the cable housing out alongside the starboard side of your engine, as straight as possible. Then push your core through. Once the core is all the way through, you are good to route the whole assembly up to the linkage.
If the solid end is a bit bent, straighten it out and file out any marks you make on it with pliers or tooling. Try again.
BT Docs pic is the correct routing. That part is pretty easy.
Best of luck. You'll get it.
Rick