It also says in Sefer HaChinuch perek 239 that you should give someone tochacha privately and in a nice way; but if they don’t listen to you, then you should embarrass them in public so that they will do teshuvah.
This truly is bad advice . Embarrasing someone in public may very well backfire. What is more dangerous , A lion or a wounded Lion. A wounded Lion can fire back at you especially with the internet . You can fire back very cheaply and get lots of coverage, Likely more coverage than whatever transgression was committed.