Sites feel pressured into Discord because it is popular, leaving bad experiences to build resentment
I once heard a person say that they dislike stats so much that they will never join a site that has them; they went on to say that stats affected nothing in roleplay, were a waste of time, and were meaningless math. In a different time, stats were none of these things, but efforts to cater to -lovers and -haters combined created a common system where numbers exist, are work, but have no value and affect nothing. No doubt, this person had a bad experience because they tried something even the site's admin's likely didn't understand.
This is the exact same trend as Discord. The first thing every site admin hears about discord is "Do you have a Discord?", and then they feel pressured to get one because everyone has one, usually not realizing that Discord has its pros and cons.
I firmly believe this fad will continue the way of stats and people will eventually begin to dislike it because they don't understand. Unless Discord can fix the cons...
Guests don't see activity
Discord is a separate app that has to be launched and there really isn't any integration for a real-time chat on a webpage. There is a brand new widget for websites where you can see who's online, but that doesn't say much as they may be bots or people AFK. So, whenever people do things on Discord, a guest doesn't see that. All they see is what's on the website. In fact, in order to see, they have to launch a whole new app, figure out what the discord join URL is, choose a handle, and log on. It's not very guest visible, leaving a guest with more "Is this site active?" than before.
Guest questions don't get responded to as quickly
I don't know any staff who monitor both the Discord and whatever cbox/chatango/etc there is at the same time, especially while idle. Discord has a way of sucking up your attention, meaning that responses to guests take longer than usual. This also plays into how welcome a guest feels at the site, and their interest usually hinges on whatever the question was.
Most roleplayers have very little in common
Perhaps the most important point about Discord is that it creates a community for like-minded people. While in Discord, chat turns to personal lives, media, music, and video games pretty often. Roleplays have encouraged players to be a community for a very long time, but the truth is that most roleplayers just don't have that much in common, which can mean that even the most valued roleplay members are suddenly the least valued in Discord, falling through the cracks.
Activity tapers off on the forum because it encourages players to plot but not play
This is the one most important reason that Discord can be bad for a roleplay. It sucks up your attention and people go off to chat about other things. This is nothing new about chat, but now that the roleplay is no longer even on your screen and that you're surrounded by your buddies from other places inside the app, the temptation to wander is much stronger. Personally I feel that plotting has gone up due to players being more accessible, but actually playing the game is much less for forum-based roleplays.