You're living in a dream world if you think you're getting away with your privacy in any way.
Also, the IRC system IS actually good, even though there's still no privacy at all.
I'm just ignoring the privacy part of the discussion.
The Two-way Auth is just another level of security, as there are always people getting hacked (most of the time it are mainly weak passwords.)
But from time to time a database gets hacked and passwords leaked.
I know 2 persons whos Steam/Twitch accounts got "hacked", and someone changed the password of my facebook gaming account (he had a fucking unsecure password)
Pretty much everything happened within a few hours. So there is a chance that there is currently somewhere a leak.
Knowing that there are people whos Steam inventory is worth 5 to 7-figures (most likely distributed over seperated accounts) a Two-way Auth is a must-have feature. (atleast as option)