EDIT: I though you are replying to the comment about just hosting single-user instances, and assumed that you meant that if everyone had their own single use private instances, it would be against the fediverse idea. Sorry about that.

I wouldn't say that's making the fediverse private - it's only making my personal account and data about what I visit private. That's what the ActivityPub protocol is for, and the more I think about it, the more I hope that some kind of app would show up - one that would be designed to just act as a personal front-end for the Fediverse, which would allow you to interact as a user from your instance with others, but also one that would keep all of your data, which are currently at mercy of your instance admins, at your personal instance.

Of course, you still need people to host instances that are actually made for communities and content, and that's what Lemmy or Mastodon is designed for - but I'd like to see a Fediverse app that isn't made for hosting content, but only for letting you interact with other instances. There's no drawback - quite the contrary, instance admins don't have to deal with and take care of my private data, because my instance is handling all of that, while I still will be providing content for their instance. I think that definitely fits into the idea of what Fediverse should be.

The only thing I'm not sure about yet is if it's possible - if I create a Post on an instance that's not my home, who is hosting the data? Do I only send ActivityPub Create Post with the data and the instance then saves it, or do I create the post on my own instance, send an ID, and if someone requests the Post data on the instance I posted to, it will be requested from mine? Because if it's the first one, then such a client that only implements DMs, your own user account, and a frontend for showing posts on other instances would be doable. And definitely something important, because it solves the biggest privacy issues of Lemmy right now. I see no drawback in that - the only data I would not be in control of are the ones I post to other instances, but that's ok. And even if you would be the one hosting it, all it means is that it would be a little bit harder do host it yourself.

Also, if I understand the ActivityPub right, if you're ok with not getting notifications or DMs, your personal instance wouldn't even need to be online at all times, since you only request data about communities and posts when you are browsing. But this would depend on whether the content and comments are hosted at your instance, or at the instance you are commenting or posting to.

I really like this idea. And from what I've seen of the ActivityPub protocol, it should even be that hard, aside from the UI.