Ultimately, I find it incredibly harmful to tell people who are concerned about climate change that the best thing they can do is to not have children. In fact, the best thing they can do is to have children and to inculcate them with a sense of responsibility for the environment. Bring them up to be minimalist, car-free vegetarians. Nurture their interest in science. Combating climate change requires smart, motivated young people to create new technologies and to normalize low-carbon living. There’s lots of evidence that your biological children will grow up to be a lot like you. If only people who don’t care about climate change have children — well, you do the math.
You have accurately observed that any solution to a large-scale problem that doesn’t begin with “take money out of deleterious system X and divert it into education” is likely incorrect. The solution you describe is a better-educated populace. A better-educated populace has the twin virtues of a sustainable fertility rate and a wide range of people who can understand the above arguments.