Its would be easier to understand how the education system is structured and designed, to comprehend why there is not aggressive competition for enrollment. Firstly, over 85% of students go to public schools funded by the government. The rationale is that the parents paid the taxes, they might as well make use of the education.
Public schools are known as "Folkeskole", which means municipal schools. Unlike many countries, primary and lower secondary education is consolidated in one school, so there is only a need to access a student's capability once and enroll the students once. The exception is when they move out of the municipal and have to relocate schools of course.
Rankings of schools are not perceived as the priority for parents. I would say, those who are very picky, tend to be the affluent who sends their children to private schools (which advertises themselves based on admission rates of graduates to different universities). So how do parents make sure that the education has quality? Each municipal board audits the school with a quality report that talks about everything from academic level, extra programs, improvements etc. Perhaps thats how parents are kind of assured that they have nothing to worry about. There is of course many academics that claim that such assessments are not comprehensive, but since the parents are not complaining, there has not been much revamp.
[Can't insert much text in this box. Source: I did an exchange program in Copenhagen Biz Sch]