So the new plan for carving the chair backs involves a router and thus a router jig.
This involves a platform for the wood to sit on and an elevated and angled guide for the router to slide across. All I have to do is drop the router to the depth I want by eyeballing the curved pencil line, and then I can sweep it across and it will conform to the desired angle.
It will still take many passes of the router to eat away enough material that I can start using sanders. But it's a big improvement over the table saw method. You can see some initial passes below, comparing the narrow table saw cuts with the wider router cuts. One way or another - neatly or sloppily, I just have to take away the material within the pencil line