Coming from someone who's friend did this...no....no they don't. Not even remotely. You just got used to your stink, and people typically aren't assholes enough to say anything about it to you. It's about as effective as using a magic crystal to ward away the smell. Hell, the salt stick even smells like an armpit after the first 2-3 times you use it.
I like how it says on the package "Apply to clean underarms. Lasts up to 24 hours!"
Translation: "Look, you filthy fucking hippie.. if you just rub this rock of salt on your stank it's not going to do shit, so don't pretend it's a substitute for bathing. You have to clean yourself every goddamned day, FFS."
It definitely doesn't work for everyone. I have known plenty of people who use them and they do not stink. I'm sorry it didn't work out for you but don't spoil the party for everyone else.