Nutrition is definitely a factor, but probably not the major one.

I think the number one factor is probably access to parental attention.

It's shocking how much better an elementary age student can do when there is a parent to remind them about a homework project that is due. Or be available to answer a question on basic math or the meaning of a word.

Even in two equally driven students, self learning gets stuck at home in ways parents can unblock if they have the time and energy to do it.