You have missed the more important aspect of a private tracker. A torrent may have been created and shared via a public tracker as well as a private tracker

You have missed the more important aspect of a private tracker. A torrent may have been created and shared via a public tracker as well as a private tracker ... however given the boring nature of some files, the number of people sharing via public trackers and even DHT (shown) can be non existent after even a few months (some torrent clients are better than others and ymmv) ... and from my experience don't do a very good job for rare or one off file shares, of discovering and then linking a seed or peer to another peer. However if a member of a private club / tracker of like minded individuals which is likely to be a smaller community, given its specialisation in a certain interest and guidelines to ensure members share many times, there's a much higher likelihood of an active file share that won't be found any where else.