You are an exception based on my own results. Of course I said forget it after a few raving lunatics.
Sorry, I still haven't heard a legitimate reason for making a tree private.
There'll always be clickologists and idiots. All anyone can do is explain (with proof) the correct lineage and ask them to correct their tree. If they do great. If not there's nothing you can do about it so time to move on.
Hiding your own tree with the correct information just helps the wrong info proliferate.
If I were building my tree based on everyone else's research (which I'm not) I'd accept the version I see most often over the version with a lesser number of matching trees.
I'm dealing with my own 3rd GGrandmother Rebecca Cotton (1826-1916) right now. All my evidence says she was born, lived and died in the Wake County NC but I'm constantly finding trees with her married and living in Alabama or Tennessee. I send the tree owners a message and point them to my tree. I also can typically cite one of the records in their own tree that would make it impossible or nearly so for my Rebecca to be their Rebecca. Some take the hint. Some don't. I don't lose any sleep over it.