My parents had 4 kids, I was the only one that was overweight. They started restricting my food (no desserts) when I was 9. They put me on a calorie restricted diet when I was 11 (I lost 25 pounds before I hit puberty, i gained it back the following year) My mom would go running with me after school. My dad coached my community soccer team.
I ended up overweight as an adult, obsessed with dieting, resulting in cycles of dieting/binging for year. It took me years of therapy to develop a relatively normal relationship with food. To this day, I weigh myself every day, track what I eat aiming toward a calorie goal, run three times a week. Yes, I'm still overweight (I dont hit my calorie goal each day, obviously), but if I dont do all these things, I gain weight quickly. Last year, I stopped weighing myself daily and tracking my food and I gained 30 pounds in 6 months for example. I dont seem to have an internal mechanism that indicates for me when I've eaten then right amount of food. I believe, though have no proof, that my internal mechanism was broken by too early and too intense dieting.
I now have 2 sons, one is normal weight one is overweight. I cook healthy homemade meals every day. We all run together 3 times per week. I am involved in both their lives. I have always emphasized paying attention to your bodies and stopping eating when you are full. I never make them, "clean their plates" etc.
With that as context, my problem with your argument is (1) parents have limited control over their child's weight, as evidenced by the same home environment resulting in some kids being normal weight, some being overweight. (2) Trying to change a child's weight can have long term negative consequences for the child.(3) something like 95 percent of people who try to lose weight are unsuccessful in the long term, what makes you think parents can achieve a better outcome than that for their kids?