As a discord user, it's the kind of platform that I would want to have running to receive notifications, sort of like the SMS of gaming.

A large part of my friend group use discord as the primary method of communication, even in an in person context (was at a festival a few months ago with a friend, and we would send texts over discord if we got split up) so maybe its not a common use case.

It leads to me dreading having to start it (or accidentally starting it - remember IE?) and opting for the browser instead.

I strongly doubt that!