In reviewing the posts here and thinking about this I think I'll vote in favor of keeping the private game function.
I believe the most compelling reason that has been articulated is for folks who want to participate in games without those games appear on various lists (in the user profile, etc..). One example scenario is for a group of RL friends who want to have a game here on M-W without using the advertising functionality. Other scenarios seem to include folks who simply want to make sure that everything in the game is "players only" without having to worry that perhaps they might forget to check the "private" box when creating a thread.
I personally think both of those are valid reasons to want to have private feature.
It doesn't appear (I've not read every word on this thread, so I might have missed it) that private threads accomplish the same goal, since game membership is still public in a public game with private threads. It also doesn't provide the simplification of not having to remember "private" on a thread-by-thread basis.
If I am reading this right the concern that the admins raised was one that had to do with advertising and private threads. Would it be possible or advisable to simply make the advertising feature and private threads mutually exclusive. I.e. you can have a private game but you can't advertise it when it is private and you can advertise a game but if you do it can't be private? I'm not sure if that would be any easier -- but it seems like it might allow the admins to not have to do a major redesign while preserving the private game functionality.
I've played in a lot of games on the weave, GM'd a few. Most of them are public and some of them include both private and public threads. I did create a private game here and liked that it was private. Ultimately, I did choose to make it public (when I wanted to go recruiting new users). However, I liked the idea that I could create it as a private game.
Would the removal of the feature be a "deal breaker" for me. No. I'm good either way, but for the record I think private games are a nifty feature that does have some real utility.
Finally, I really do appreciate the hard work that the admins put in. You all do a fantastic job and we don't tell you that we appreciate it nearly enough!