I think your order is wrong. If a math teacher had an entire classroom of students that couldn't add but were great citizens we would probably say they failed at their job. That math teacher or English teacher has 1 job at the end of the day, (no matter how many tasks an administrator assigns along the way). If the kid can't do math or can't read/write then the teacher has failed, (or maybe the kid but for sake of this topic we are talking about teachers and their job). In reality, parents, community, church, and family teach citizenship. teachers teach content, (and I think that has gotten lost to some as teachers see themselves as the most responsible adult in any child's life). We teach music, specifically how to play an instrument. By-products of playing in an ensemble include productivity and such.
We would do well to remember to teach the students our content and let the parents raise their children even if we don't agree with how they are doing it. Stay in our lane.