> the government [...] only know[s] that I needed to verify an assertion about my age

This is problematic if a majority of things needing age verification are looked down upon; for example, insurance companies would love to know what people don't do things needing age and therefore don't buy alcohol (at least not online).

The first question is how would the insurance find out that you are doing lots of things requiring age verification? The only body that could tell them is the government, while a distrust in the government can be healthy, I think this is the least thing to worry about, the government typically knows already much more damaging things than how often you ask for age verification.

Moreover, that would only work if there are relatively few things that require age verification and it needs more than just being looked down upon, i.e. while alcohol buying might be interesting information for insurances, watching porn is likely less interesting. Even worse, if the insurance can't distinguish between porn and alcohol (which they can't by design even if the government would give them the information about how often you ask for age verification).