How to patch small anchor holes in plaster wall and potentially reinstall anchors?
Since this is just a towel rack, not a catwalk in a power plant or something like that, I see no harm in tinkering a bit.  

Very easy fix, use small zinc auger type anchors, they make a bigger hole in the drywall for a smaller screw, so these should work in the existing holes:  



You want the zinc ones since you might want to unscrew this at some time.