Given the increasingly inverse educational outcomes relative to per pupil spending, it really does sound like they've reduced a complex situation down to a single variable that has a rather poor predicative function.
Different areas have different CoL. So if you compare unadjusted spending per pupil you'll get a skewed picture. Different areas also have different mixes of cultures, expectations, and parental challenges like poverty or needing to work multiple jobs. The article specifically calls out that families in our capitalist nightmare need help feeding their kids and caring for those too young for school.