Doc Green
I've turned a lot of rolling pins (big, straight spindles with handles) and my go-to tool when I get near the final diameter is a scraper of some sort. The scraper might be the wing of a bowl gouge, a traditional scraper used in the shear scraping mode, or a round-nose scraper with an included angle of about 35 degrees.

The thing that led me to scrapers was . . . the ripples, as described above.

No matter which type of scraper I use, it cuts and makes shavings as opposed to simply scraping (abraiding) the wood away. This requires proper sharpening, of course.

For refining the surface of a spindle, shear scraping with a traditional scraper with a proper burr made with a diamond hone is hard to beat.

An excellent tool is a scraper with an included angle of about 35 degrees sharpened to a knife edge with no burr. It will cut, make shavings, and leave a good surface. Further, the depth of cut is pressure sensitive - press harder for a deeper cut. Or, use a very light pressure to remove just a wisp of wood.

This tool is described about half-way through this article:

https://www.docgreenwoodturner.com/scrapers3.html

The details of shear scraping and sharpening a traditional scraper are covered in two articles prior to the one linked above.

Good Luck!