Finland's schools are really good for a number of reasons, I'm not sure that private vs public is the only reason worth attributing it to, although i understand the context of the article makes it especially relevant.
For example, Finland provides three years of maternity leave and subsidized day care to parents, and preschool for all 5-year-olds, where the emphasis is on play and socializing. The state subsidizes parents, paying per month for every child until age 17. 97%* of 6-year-olds attend public preschool, where children begin some academics. Schools provide food, medical care, counseling and taxi service if needed. Student health care is free.
(* a decade ago, not sure if numbers and strategies are still accurate, I lifted it from a Smithsonian article from 2011 because I couldn't remember specifics. Please correct me Suomi friends)