That seems intuitive, but it's not actually true. I suggest looking up zero-knowledge proofs.
Using modern cryptography, it is easy to send a machine-generated proof to your social media provider that your government-provided ID says your age is ≥ 16, without revealing anything else about you to the service provider (not even your age), and without having to communicate with the government either.
The government doesn't learn which web sites you visit, and the web sites don't learn anything about you other than you are certified to be age ≥ 16. The proofs are unique to each site, so web sites can't use them to collude with each other.
That kind of "smart ID" doesn't have to be with the government, although that's often a natural starting point for ID information. There are methods which do the same based on a consensus of people and entities that know you, for example. That might be better from a human rights perspective, given how many people do not have citizenship rights.
> (and probably unconstitutional under the First Amendment, to boot.)
If it would be unconstitutional to require identity-revealing or age-revealing ID checks for social media, that's all the more reason to investigate modern technical solutions we have to those problems.