I think it's not that people want a forum and choose Discord to host it.
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.