Of course I have an answer. To do something about the unlimited firehose of porn, violence, divisive, and addictive content that has been pointed at children for the past generation or so.
There's literally nothing confusing about the "why" in this discussion.
The fact that bad people use the "what about the children" argument regularly for bad reasons doesn't mean that all such arguments are bogus.
In fact, it's an indication of exactly the opposite, it's so regularly used because there is a broad consensus that we need to protect children from harm which is why it's often effective as an arguing tool.
The relevant frame for this discussion is will it actually work, and what are the tradeoffs. A trivially small amount of money for a simple age verification scheme isn't a particularly meaningful tradeoff against a genuine social problem. The bigger, more genuine issues are around privacy and censorship and I do in fact concede those are real.
See, you've answered a different question. The question you answered is, "why should children be protected from the Internet?" I'll give you for free that children should be protected from the Internet, for the reasons you've said, and now you get to convince me that I, who don't have or plan to have children, should spend my money to protect other people's children from the Internet.