LLMs are not "someone", LLMs are something, and they don't "read content", they by definition acquire and reuse that content (for example, by summarizing it), as part of their product.
So here the consent is indeed about what can be done with the data.
In general, it's absolutely the norm that public websites (I.e., unauthenticated) restrict even who can access the data. The simplest example that comes to mind is geoblocking. I have all the rights to say that my website is not made available to anybody in the US, for example. Would you still call that website "public"? Would bypassing the block via a VPN be a violation of my consent? This is mostly a moral discussion I suppose.
But anyway, it's not what's happening here. LLMs access content for the sole purpose of doing something with that content, either training or providing the service to their customers. They are not humans, they are not consumers, they don't simply fetch the content and present it to the users (a much more neutral action, like curl or the browser does). It's impossible to distinguish, in the case of LLMs the act of accessing and the act of using, so the difference you make doesn't apply in my opinion.
