I was an extremely reluctant convert to Discord, at first. One of my mods was actually the one to take the plunge and create it, so I'm not even the owner of the discord channel. xD We started to use it only because the cbox was blocked on the mod's work PC. (Not the forum, just the cbox. It was weird.)
I've grown to love it, mostly because the mobile functionality is amazing compared to cbox's god awful mobile integration, and it functions as a private chat service as well, which cbox doesn't offer. We have some bots (namely Welcomer bot) that help out a lot as well. But I was really reluctant for the reasons you described. Moving chat off site and into DMs on what's supposed to be a writing forum, I was really leery that activity would start to move off site (both in terms of chatting and plotting in the plot sections).
And it did, but the impact was actually fairly minimal? We had issues very early on with 'secret' groups and separate servers players were creating that was leading to a lot of confusion and hurt feelings about other players feeling left out or that those secret groups and private servers were forming their own cliques. We fixed it by offering both public and private subplot channels dedicated to their plots and we've had no issues since then - and all staff have access in case something is going on that involves moderation.
People are going to move into a chat service anyway, like Discord, Slack, Hangouts, Skype, whatever replaced AIM, whether you offer them a place to do it or not. At least with a site discord we (as staff) can have at least some eyes on the general conversation. People who prefer instant communication over plotters and PMs are still going to go into DMs or into their preferred chat of choice (mine was Hangouts, before Discord was a thing) but at least offering them a site discord to use 1.) still keeps activity somewhat connected to the site, and 2.) gives players a space for the instant IM chatting.
The whole 'external' bit is definitely a problem - we still keep our cbox on the site and will likely never get rid of it completely because the forum integration is the one thing cbox can still lord over Discord with ease, I hate Titan - and the guest options leave a lot to be desired. But Discord is the closest chat service I've seen in a long time that offers both a chat service and a community hub-like feel. If it fixed the external integration issues and allowed guest options in a cbox-like manner? I could see cbox fading from forums completely.