If you also want to host or build interesting social apps, you should definitely do an isolated atproto / Bluesky service! https://blueskydirectory.com/
As for actually doing this... running a PDS and relay isn't that hard, and the red dwarf web client is online and can be configured to point to whatever appview you want. There's significantly less experience running your own appview, but there are options & folks are happy to help.
I’ve used bluesky, and it’s very twitter-like. That doesn’t seem like the best model for a close-knit community. For larger ones, perhaps!
That you phrase it like that implies that you haven't used atproto very deeply, and aren't aware of how versatile the protocol is, and how many apps it hosts.
I linked you a directory of apps already! You could use any of these! You'd have to set up your own instances to use it on a private service but that's doable, and since you'd have the main atproto systems up, it would be much lower lift than you might expect!
PDSls let's you browse people's PDS. This shows you what apps I've used! It's quite versatile, capable of hosting all manner of social systems. There's nothing else that will give you the ability to build a neat rich social community like this: everything else has specific purpose and intent, and you are rather stuck with that design, but atproto is versatile and generic and ready to form whatever kind of social systems you want with it. To look at what's here and say atproto is very twitter like is to barely scratch the surface. https://pdsls.dev/at://jauntywk.bsky.social
There are projects that make running independent atproto networks "easy": https://github.com/verdverm/testnet
I no longer recommend ATProto, in part because the public by default was a terrible choice. People prefer privacy, not anyone in the world able to read all of their activity. Bolting permissioned buckets on after the fact is not the way, it needs to be core to the protocol design.
I just started looking at the At Protocol for another side project - do you think the protocol will eventually support such privacy settings by default, or is heading in that direction?
It's baked in as deep as it can go.
Use a different protocol.
Read the "perm'd data diaries": https://dholms.leaflet.pub/3mhj6bcqats2o - It sounds like they have already picked this rough sketch as the basis.
My take is that (1) public vs private will be an app level choice, and user if the app passes that choice through and (2) this sketch is insufficient for many applications, being on the simpler side of the design spectrum.