yoshi, Icouldn’t take that as a challange! 🙂 All I did was quote Sefer HaChinuch, not offer any chiddushim of my own.
You are correct, about not embarrassing someone. But (as a point of comparison) the Torah also says you shouldn’t kill someone (one of the 10 commandments.) Yet, under certain circumstances (i.e. someone who violates Shabbos), the Torah demands that we kill such a person.
Here too. The Sefer HaChinuch says that you should give tochacha privately and with love. If after exhausting all such attempts the sinner remains obstinate and refuses to mend their ways, the Sefer HaChinuch says you must publicly embarrass the sinner in the hopes that will prompt them to repent.
(Again, this is not my own sevora.)