The argument I've seen for forums is people do not want to deal with the burden of hosting and securing forum software. But that is an explanation for reddit eating most forums to me, not why Discord is increasingly being used to place of them.
Instead it's that people have a Discord for hosting community discussion (like one would have an irc), and they add channels like "q&a" or "show and tell". These channels are much more topic based than normal channels, and topics collide or get lost in the noise.
I do think public indexability is a big problem. I've seen a couple api based bridges that have surface, which are great. Discord's change to how names work smells strongly of a move to being more aligned to a social network, so perhaps they're already working towards doing this properly. It's a natural evolution from chat to forum channels to public indexed forums, but not one that was immediately obvious when discord started as a game with your friends product.
Now that I'm typing this out, I just realized that "subreddit" style channels in discord make a lot of sense, where you can get an aggregated view across all of the "subreddits" in the servers you're in.
Discord to function like an IRC chat and a Wiki to store the important information on the topic of the forum.