How do you prove your age without either sharing your identity (or using twisted ways entailing other possibly unacceptable compromises like the eu is trying to do).

Remember that the australian side (which moved first) framed it as "no underage access to encyclopedias, no underage access to search engines": age verification, i.e. identity verification or other "nonsense" to access encyclopedias and search engines. In the USA, some voices would include LLMs. Remember that in the uk Wikipedia had to win against the state to get exception under the mandate from keir starmer - it "exceptionally won", and after having lost at a first stage.

The privacy of online browsing and the privacy of messaging are equally critical, yes.

(Edit: if you wish to really rank them, then a guy looking behind your shoulders when you are reading a book is more dystopian and farcical than a guy looking behind your shoulders when you are writing a letter - yes.)

You don't prove it. Under California AB1043 (often called the "age verification law" by propagandists) it's illegal for a social media website to demand you to prove your age.