As a coach, I teach at both after-school clubs and at my own chess school. As a parent, I send my children to both after-school clubs and some classes outside school. I can tell you there are big difference between the two.
First, different student groups. At after-school clubs, the students are (in)volunteering to attend the clubs. The parents have difficulty to pick the children, so they just want to find some interesting or meaningful activity for their children. The children may or may not like chess. They have very little motivation to improve. For classes at chess school, it's the opposite. The students are self-selected. They want to learn and they want to improve. If your child wants to learn chess, which environment you want him/her in?
Second, most after-school clubs are just one quarter or at most one year. The teachers know their assignments are just 6, 8, or 10 weeks. They don't have long term vision. It's just a short term assignment. So they don't have any plan. Every quarter, they just repeat the material. They don't and can't care much about students' progress.
Third, when everyone has very low target, students have little motivation, and teachers are just fulfilling the contract, what can we expect? Most after-school clubs will always have beginners. Anyone who gets better will leave the club, because they don't want to waste any time anymore. This is a reality we have to face. Until the school is treating chess as part of curriculum, our children have to move on their chess study to outside chess schools.