A rasp and lots of sanding would work. Professionals would probably use a drawknife and spokeshave; a YouTube video, “Spindle Repair on a Chair Back” by Fixing Furniture, shows how.

With the sanding completed, dry assemble the chair to check the fit of all the parts. We used Titebond Extend wood glue for the spindle assembly because of its long open time. First, apply glue ...