Instead, there should be an HTTP header indicating that the server is an adult-only website. Then, child-locked devices can refuse to show the content to their users. Moreover, this can be more granular than just a single adult-only bit.
This standard has existed for 30 years and was supported in Internet Explorer 3; it's called PICS: https://en.wikipedia.org/wiki/Platform_for_Internet_Content_...
If the current age verification controversy was intended to protect children rather than destroy anonymous speech, it would be focused on requiring the implementation of PICS or something similar.
what happens when the request goes through a proxy and that proxy is configured to strip this header?
All the age verification is is creeping totalitarianism by governments.
No, that leaves vulnerable adults unprotected. It should be "Is my thing in one of the categories of things that this device says this user is not permitted to do? If so, I shall not permit this user to do the thing.".
Nothing stops software authors from providing pre-built bundles of categories that they believe fit certain types of vulnerable people [0], but the fine-grained control must be there so that guardians can choose to set things up for those they guard so to adequately protect them while minimizing the amount of stuff that they're blocked from.
[0] Like: "Overly-trusting human who needs protection from scams", "Dementia-damaged adult who cannot be trusted to manage their finances", "Median sixteen year old USian", etc, etc.