Who is suggesting people 'show their papers' to go into a public square?
Literally nobody.
It really demonstrates how bad the analogy is - so much so that it's not even analogy.
The 'social controls' on the 'public square' are limited by a few laws (aka directed violence) but apart from that you can say as you like, kids can as well - it's where parents can be parents.
And - don't have problem with kids in the public square.
We have a very real problem with kids on social media, verifiable, scientific.
Kids are depressed, distracted, they bully each other, they're creeped on, and they're not yet in the business having serious discussions about 'Mein Kampf' - they're kids.
Everything in kids lives is introduced in an 'age appropriate' fashion - literally everything.
Given the toxicity of social media, it's a 'primary concern' for one of those gated things.
This is not even an argument - the only argument is 'the slippery slope'.
If the point were to improve on the mental health of kids there are countless underfunded public programs. Especially in the US, social support programs like food, healthcare basic and mental, actual physical public spaces for kids, arts in curriculum, etc.
Conversely, I don’t agree with the way some other countries are going about it. Especially the UK with the abysmal way they have physically policed online speech by adults. Incredibly sad to see police prioritise non-violent “speech crimes” because they’re too scaredy-cat to tackle actual violent crime.
There is a reasonable answer to be found, if we're willing to be inventive. It shouldn’t be beyond the imagination of cryptography experts to design a system where only governments can issue an age identification certificate, which individuals can use to generate verifiable proof of age tokens. But where the tokens can’t be used by the government to identify the individual.
It should even be possible to make a protocol where the user who proves their age can also tell the verifier to block any future requests for a day or so; and in such a way that it's impossible to sell one ability (proving your age) without selling the other (being able to block yourself from proving your age for a while).
Such a protocol would make it risky to sell age verification to other people, because a troll could block a seller's ability to make more age verification tokens. It would deter selling while still keeping the government from knowing what services you're using (and the services from knowing who you are).
The government shouldn't infringe on my right to bear arms, mass shooting is a mental health issue anyways, oh but we can't really fund mental health support because my tax shouldn't go towards helping those who put themselves in that situation. Que the same argument for privacy on the internet.
You can't have your cake and eat it too. If you can't trust the government, then work towards healing it, restructuring it, overhauling it. Subverting the government is such antisocial behaviour, very criminal like.
I actually like the traditional public square model: you have anonymity most of the time, but it’s not some absolute shield you can abuse to be an obnoxious prick. The police can intervene, but the intervention happens in the public square too.