We use IRC for random chat stuff currently. It's old but reliable, and we help to run the network behind it, so we know it can be trusted.
I'm not entirely against adding new ways to communicate, but my main concerns with Discord would be:
- I really prefer to use our own servers and bandwidth, giving us as much control as possible. Discord uses servers that others entirely control.
- Discord is still quite new and losing money, with a business model that is very much in flux. I'm not sure where they'll go with it. Skype is an example of the direction they could take (selling the company or adding plenty of advertisements), but they could also start going other ways, such as monetizing data mined from user chats. I hesitate to encourage the use of something that could be changing significantly in negative ways.
- We have to think carefully about adding more media that staff have to keep tabs on as part of the job. There's already a lot they have to do!
- Staff here shouldn't be chatting with customers by voice

. Voice-based communication is really inefficient in general.
There's an element of "flavor of the week" to Discord, as it's the latest cool, shiny chat software for gamers (IRC is a super-old, uncool thing, and Skype, Slack, and the standard rented voice chat programs are also no longer cool). But, Discord also has improvements that other software packages don't have, and significant momentum within the gaming community thanks to widespread, enthusiastic boosting. Those could end up giving it real staying power. Think Facebook (which was a late iteration of traditional social networks that had these factors over MySpace, Friendster, and the like) and CloudFlare (which is a recent iteration of CDNs and has these factors over old-school ones like Akamai -- with tons of marketing to boot). It will be interesting to see where it goes.