I mean, you can run your own frontend if you want to? If you can run a backend, you can run a frontend. You can’t get away with hosting a relay or an app view, though, you need to host a full personal data server, with your own indexers and everything. You can also rely on external indexing, but then you turn dependent on the servers you interact with.
As for federation, you can federate with smaller services just fine, but Bluesky’s design makes it very centralised around the main server. There’s nothing on the protocol level that enforces federation with the main site.
This is a bit like with the Fediverse: for a couple of years, the only relevant service on there was Mastodon, specifically a few larger Mastodon servers. If you didn’t federate with them, you may as well not have federated at all. This also made some of Mastodon’s weird quirks de facto standards for Fediverse support (like how some ActivityPub object types aren’t supported, and people put every huge web page in a “post” object so it would be rendered by Mastodon).
If the main Bluesky server bans you, there’s nothing preventing other Bluesky instances from indexing your content just as easily.
If anything, this setup allows for more freedom than (for instance) Lemmy does. If Lemmy.world bans you, you can’t post in their communities anymore. Other servers can’t discover your attempts to post, so your local posts will remain local, out of reach of any ActivityPub server. Sure someone could scan the Lemmy API for differing posts, but that’s not how Federation is implemented here.
On Bluesky, independent servers can (should) index each other using the standard API. You can get banned from bsky.app, but other servers can still see your posts pop up in their feed.
I don’t know for sure how banning works on a technical level, but it is my understanding that moderation on Bluesky is actually done not dissimilarly to nostr. Everything gets processed and stored, but there’s a layer between the database and the frontends that handles blocking and banning. Of course Bluesky could disable federation with your server entirely, but it seems to me like they’d still federate your content even if your account is banned using normal banning tools, unless other servers propagate your ban as well. That’s up to them to decide. This too is unlike ActivityPub, where an account ban will also halt propagation.
From what I can tell, it’s perfectly possible to set up an independent Bluesky network. The only issue is that Bluesky users don’t give a shit about federation and won’t move to any alternative server, so if you want to reach them, you need to play nice with the main instance.