The main issue is that they are very careful not to frame it like that. In broader contexts, it's always framed as something like "do you favor limiting children's access to social media" without a word on what it would cost to actually institute such a ban.

Yeah, and we’re starting to inoculate people against that kind of rhetoric. It’s a process.

Relevant British comedy clip (Yes, [Prime] Minister) on such polls:

https://m.youtube.com/watch?v=ahgjEjJkZks

It's about as meaningful a framing as asking if you favor world peace and ending world hunger.

You don't necessarily have to be in favor of any measures which reduce adult privacy to be in favor of that. Logically speaking, the liability for minors accessing age-gated products and services is the person who provides those products and services to the minor. In the case of the internet, that person is the parent, not the ISP or the website. It is the parent who contracts with the provider and then forwards the product to the unauthorized user, the child. A parent who purchases, say, access to porn and then provides that access to their child is no different than a parent who buys booze and provides access to it to their child.