There was a time when "invite only" would've worked. I was a member of a gaming forum that did not allow public viewing. It eventually died off a slow, slithering death. Because the owner didn't care for the drama that came at the peak of the site. A sister site was built, that also died. I was the one who insta-killed the community. Not my intent, but whatever. The idiot left the database for download - for free. I took the opportunity, and ran a few tests with it. Bunch of haters got butt hurt and tried to hurt my name. *wipes off shoulders*
The moral of the story is that there was a time when this was okay. Today, not so much. I think people got tired of the business model - and not just from the admin side of things, but also from the user side of things. They want to see what kind of content is there before committing to the "cause" (if you know what I mean). And then, there's the case of "he said, she said." You can screencap it, but then you can photoshop it... So, you'd have to be on the board to see if the person did it or not.
Another example is a board with 3 Million posts, had a single forum private. It was kind-of a "adults only" forum, where you can talk shit about anyone, joke around, or whatever. One day, we had like 30 users online to a single thread because everyone was trying to find out what I'd say next, or find out my ID. At the time, people thought I was not 21 (to which I was), just too many people assuming, too many skeptic, too many losers just spreading rumors about me. This was at the height of AOL, MSN, and other chat services, too.
If there was not a single popular person that could make viral rumors, speculation, skepticism, etc... that forum would have just stayed the way it was.
You need to make that private forum extremely popular, extremely controversial, or the private forum will just fail. Are you going to put up with the drama that comes with it? I think not.
The only private forum I could think of that would work in today's market is what we have now: ResetEra and NeoGAF. Both are not "private" like you want, but bear with me: NeoGAF used to be a popular place to post, but it did have a lot of bad apples. Once the forum had a few mods that were shrouded in controversy (the bad kind of controversy - not the kind of controversy that attracts you to places, like the above example), there was an exodus. Mods followed suit. Because those mods were pedos. The next controversy was the owner, whom was accused of sexually assaulting a lady. (Gamergate/MeToo) There was a much, much larger exodus, but this time it was the best thing for the forum. Most of the people that left NeoGAF, collectively created ResetEra, members flocked to it. But the bad news is that the forum is more of a left-leaning hivemind. It's actually a "mini-cancel culture" based forum. So, if you don't meet them eye-to-eye on many subjects, like an SJW... You're casted out as a loser. This is all public record. All out in the open, though.
A private forum for SJW's would work. A private forum for left leaning losers would work. But only to a point. There was another forum I have seen that's similar to NeoGAF/ResetEra, but it's dying a slow, very slow death. It's now a discord channel. It was never popular to begin with. They needed someone like me to come in and make it "interesting."
I ain't gonna help ya, see ya in hell, asshole.