The only thing server owners control is whether the server is public or not. They can’t set this particular aspect of it on or off, only individual users can.

As for turning a public server private or vice versa, that’s a very noticeable and impactful decision in lots of ways already.

server owners have setting toggles to opt the server out of having public facing channel history or ai channel summaries. a server can remain discoverable and have the history and summaries off for the public webpage of the server.