Out of my DCs' classmates, the ones who had a chubby toddler look around yR now just look to have a significant amount of surplus body fat in their KS2 years.
I weigh and measure mine a few times a year to check that they are "growing well" as they straddle the line at the lower end of healthy. DS1 was often weighed at dieticians due to allergies as an infant. It's not weighing that's the issue, it's value judgements made. I've always talked about food as fuel and nutrition and the jobs that different foods do. Nothing is banned, but some foods should be eaten more cautiously than others. They are not "good or bad", they just have different amounts of energy and nutrition.
Parents often don't see their children's weight, especially where overweight is normal and average. Our friends wondered why they received the letter about their children in yR and think mine are "skinny"... years later theirs have lost their energy and slowed down a lot as they've continued to get heavier and my "thin" children are faster, more energetic and have more stamina. Their lifestyle isn't far wrong but introducing a bit more intensity of exercise rather than just steady walking, swapping some snacks around and care on portion size would make a significant difference in helping them to grow into themselves with time and help them to go into adulthood with healthier habits in the future. There's no point in spelling it out until they're ready to join the dots up themselves and they only see one or two at a time.
If schools use this data to establish healthy habits such as the daily mile in a general way that benefits all, that's useful. DS has specific learning difficulties and school uses data to add in interventions to support him. We don't moan that his self esteem is being crushed because he writes below age expectations and gets extra support, we use supportive language about it (and with his combination of SENs he could be very susceptible to low self esteem)
Weighing children or not, they do realise that they/ their peers are overweight anyway and ignoring it is as damaging as handling it insensitively.