but it seems likely that the best way for me to get more people on board is to set up a family server. That way it’s not so much about “here’s a new way to communicate” but “here’s a small, private space to plan family reunions and keep everyone up-to-date on family events.”

This was a convincing argument in my case :)

Also, I have some other projects that I’ve been putting off because of the need to figure out self-hosting. Running something for the extended family might be the push I need :)

Good motivator at least! The downside/risk in your situation might be that you will expose them to quite a bit of downtime and instability initially. If you are new to this, you can probably start small, off of a raspberry pi for instance, and work out a way to always have a backup/working config on a “B” SD card while performing updates and maintenance on the “A” card (or even have two raspberries), it should help in times of doubt :) (you can just slide the other card and deal with it later)

Snikket looks like it checks off a few boxes. It looks pretty simple. It uses Docker (not that I care, exactly, but it seems that Docker is something I will have to come to terms with in general for other things on my self-hosting roadmap).

You don’t necessarily have to use docker. I deploy my own instance (of ejabberd, another server software) without docker, and have been at it for about a decade now. My (hot?) take is that it trades a tiny bit of convenience at the start for more clunky tooling and abstraction getting in the way after. An XMPP server isn’t that complicated to warrant docker (IMO). And you can get help from here, as I’m pretty sure many other admins deploy without docker and would be keen on helping you with that:
https://prosody.im/discuss#chatroom. https://chat.snikket.org/

The Android app is available through F-Droid, something that only I care about, but I do care about that.

As I wrote earlier, snikket is merely a packaged and preconfigured version of prosody & some popular clients, this is as generic and compatible an XMPP setup as you can think of. Any maintained XMPP client would work (but sticking with the ones branded as “snikket” guarantees that you won’t accidentally pick an old/unmaintained one), for instance, in my case, I use Conversations and Siskin which you can find on Google/Apple stores, just so you know those are options as well.