It's true that improving public education is not an adequate (or particularly effective) means of reducing economic inequality.
But I feel like this fact is often invoked as an argument against worrying about whether/how our schools could be improved, existing social inequities notwithstanding. And that doesn't make much sense to me for 3 reasons:
1) We should seek to cultivate kids' intellectual development as an end in itself, irrespective of the second-order consequences on our political economy.
2) Improving education can have beneficial social effects, even if it has no impact on inequality. Economic life is not wholly zero-sum. And education can help spur productivity/welfare-enhancing innovations with broad public benefits. (Though these benefits would be far greater under more egalitarian economic institutions.)
3) Kids' educational outcomes probably aren't solely determined by background economic conditions and innate ability, such that the quality of their school's administrators, teachers, curricula, and accountability mechanisms have 0 impact on the knowledge and intellectual skills/capacities they acquire.
None of which is to say that all our schools present flaws can be fixed by changes in education policy, or that the optimal set of education policies is self-evident. Just that trying to improve the schools would not be an unworthy project, even if doing so had no impact whatsoever on inequality.