> The age verification bills I've read have steep penalties for retaining information, so that seems fine since that's literally more protection than you get in person.

The best way (and only way) to prevent retaining information is to not share them in the first place.

> You can be in favor of privacy while simultaneously thinking porn, gambling, and advertisers shouldn't be targeting children.

There are other method to achieve this without mandatory identification. You can force these content to be served with an HTTP header providing their legal minimum age of consultation or type of content, and blocking them browser side. Governments could maintain filter lists for different age bracket and release them to everyone, allowing easy compliance on the device parental control settings.