About your idea, I'm not sure either way. On the surface it makes sense, and there probably is a real correlation between test performance and amount of after-school learning, so it would make sense to make homework mandatory for kids who scored low. It would also work as punishment, but at the same time reinforcing the kids' perception of homework as punishment.
What makes me really unsure, however, is parents. Families differ by the amount of time and resources they can, or are willing, to spend on child education. This is part of why I don't like the homework-as-externality model: even ignoring how students themselves feel about it, if the school is trying to maximize the amount of work they do after school, the first kids to hit the limit are ones with e.g. a single parent doing two jobs, poor household, or dysfunctional family that doesn't care. Overloading students with homework implicitly disadvantages those that don't have supporting parents with lots of free time. And those same kids will also be ones more likely to fail at tests, in which case dumping extra after-school work at them might do the opposite of the effect you intend.
But this is me speculating, I don't work in education, and I'm sure there's been research done on how to balance the amount of after-school work for the environment children live in. My complaint about homework-as-externality isn't trying to deny the work of education sciences - it's pointing out that even if the research is there and results are solid, it's not being applied anyway, because teachers are uncoordinated and they all individually think, "oh, that's just half an hour worth of work, no big deal". Almost textbook tragedy of the commons.
I resonate a lot with what you said. Homework or in the way it is used today as reinforcement work is most needed for those who have trouble picking up new concepts during limited instruction time. However, those who have trouble picking up concepts quickly will also have overlap with not having much time/resources outside of live instruction time. That just leads to a double whammy in terms of how homework further disadvantages them.
In the reading I've done, I've seen most advocate for smaller class sizes with more educators per class to help provide more 1:1 attention. This is again at odds with how public schools are funded where I am (US) so I don't know if anything will ever change.
My personal experience is homework was an excellent way to reinforce what was taught in school. For me, it felt especially useful in quantitative subjects and languages (French & Spanish) which both felt like they benefitted from having a concept stick. For qualitative subjects like writing, reading comprehension, I actually now look back and see homework as a way for teachers to see samples of your progress. "After reading this passage, did you learn how to construct a summary? Did you learn how to pull the author's main message? Did you learn how to make a convincing argument for or against the passage's main point" and I can't think of a fast way to do this in any kind of live instruction setting.
What makes me really unsure, however, is parents. Families differ by the amount of time and resources they can, or are willing, to spend on child education. This is part of why I don't like the homework-as-externality model: even ignoring how students themselves feel about it, if the school is trying to maximize the amount of work they do after school, the first kids to hit the limit are ones with e.g. a single parent doing two jobs, poor household, or dysfunctional family that doesn't care. Overloading students with homework implicitly disadvantages those that don't have supporting parents with lots of free time. And those same kids will also be ones more likely to fail at tests, in which case dumping extra after-school work at them might do the opposite of the effect you intend.
But this is me speculating, I don't work in education, and I'm sure there's been research done on how to balance the amount of after-school work for the environment children live in. My complaint about homework-as-externality isn't trying to deny the work of education sciences - it's pointing out that even if the research is there and results are solid, it's not being applied anyway, because teachers are uncoordinated and they all individually think, "oh, that's just half an hour worth of work, no big deal". Almost textbook tragedy of the commons.