There’s a Patreon for it that pays the server bills despite it only being a few hundred users. My users even say thanks for running the place now and then. Running a small node of The People’s Glorious Social Network is a very different task than what you are outlining.
How do you vet applicants? Depending on the theme of the site, this seems like it might range from easy (gardening, cooking, ...) to excruciating (politics, medical).
A lot often is just a simple matter of "if I don't like you, I won't give you access, or retrospectively kick you off". And the other side is that if you, as user don't like that, there are thousands of other instances to choose from. And if none are good enough for you, you can run your own.
But that simply doesn't scale to the level of services like Twitter. You might argue, and I would agree with you, that maybe we'd be better off without services that are too large to moderate in any meaningful way, but we are where we are. An invitation-only Mastodon network is not a viable alternative to Twitter.
It's not the network that is invitation-only. Just some servers on that network that are. The network itself is entirely permission-less.
You're not building social network then. Just a single isolated place. So obviously problems of running bigger social space won't apply as much