This is what terrifies me about Discord, as well as the fact they store all their logs in plaintext. AFAIK it is impossible to delete logs in Discord and any new user joining a channel gets access to all the back history of chat, so it's even worse than IRC.
I actually think that is fine; private messages and private channels should not have public logs, but public channels should have public logs in plaintext. However, I think IRC is better than Discord, and it is possible in IRC to have server-side logging if the server software implements this (as far as I know, mine is the only one that does), without altering the protocol. But making logs with a special client is also possible (also without altering the protocol) and seems to be much more common in my experience.
Almost every IRC channel I've ever joined had users who logged the chat 24/7. Sure, it wasn't centralized but at no point is there any tacit agreement that public chat channel isn't being logged. I don't really see a difference in this and Discord giving the entire history of the channel the user is joining.
