I lt feels like there's two separate issues: whether children should be allowed on social media, and how age verification should be implemented.
It's perfectly reasonable to be concerned about privacy, data collection, and mission creep. Those are legitimate concerns that any age-verification scheme should address. But it doesn't follow that because some verification methods are bad, we should abandon the goal of keeping children off social media.
The fact that platforms already prohibit under-13s isn't much of an argument when everyone knows those rules are routinely ignored. If a law is unenforced and ineffective, pointing to its existence doesn't solve the problem.
You also seem to assume that age verification necessarily means handing social media companies copies of passports and driving licences. It doesn't have to. There are privacy-preserving approaches where a third party verifies age and only returns a yes/no answer. Whether those systems are good enough is a fair debate, but it's not accurate to suggest that the only option is mass identity collection by Facebook and similar companies.
I also don't think it's fair to characterise supporters of these measures as advocating government overreach or surveillance. Many parents simply look at the evidence around addictive design, bullying, self-harm content, and compulsive usage among children and conclude that some form of restriction is justified. They may be wrong about the solution, but they're not necessarily indifferent to privacy or civil liberties.
Finally, if we accept that social media companies have failed to make their products safe for children, then regulation becomes the obvious next question. Saying that platforms should design better products doesn't tell us what to do when they don't.