Adding servers to the federation list is just white-listing them. It's not actually establishing syndication in any way. You don't syndicate at the website level, you syndicate at the user level.
The way the system works is, a user on your website subscribes to a user on another website, and from that point forward the remote website starts sending your website that one user's posts, addressed to any and all users on your website who requested them. Your website then receives and stories a copy of all future posts from the remote user, and adds them to the subscribing users' feeds.
This subscription is very much like a magazine subscription. Your site does not receive the back catalogue. It does not automatically receive other magazines (users) published (hosted) by the same publisher (website). You only get what has been requested, from the point the request has been accepted onward.
There is no canonical fediverse that you can just see. It's not a centralized system, which means there's no source of truth to tap into. It's a mass, opt-in content syndication technology, where you have direct access to the locally hosted content on the single website you are using. The fact that much of that content originated elsewhere presents the illusion of some centralized whole, but it's just that: an illusion.