If you were introduced to Discord in a non-gaming-related context, it is very common to be unfamiliar with the tendency of gamers to share what game they are currently playing. There is no such habit in other, more "serious" contexts - for example MS Teams does not set "currently using Microsoft Powerpoint" or "currently coding in VS Code" as my publicly visible status at work. I can thus fully understand that a lot of people are (negatively) surprised by this - even gamers, because there's a huge number of people preferring singleplayer games out there that don't have the need to do any group coordination for which a tool like Discord (or built-in community features of games or game launchers) would be needed.
Hence if Discord aims at expanding its user base beyond the (multiplayer) gaming community, it should definitely think twice whether to keep this sharing setting enabled by default. At a minimum I would expect them to try to deduce somehow whether you are using Discord as a gamer or in a non-gaming-related setting, and use the suitable default. Or maybe just disable it by default, so gamers who want this feature enabled need to turn it on.
I'm not sure why they don't just create a business version branded separately from their gaming product. Even if it's just 'Discord for Business' a second client for business would solve almost all these problems. It's written in Electron, there wouldn't be millions of lines of code to change to enable a second branding and different ports.
I don't think these types of status updates (even automated ones) even originate in gaming contexts. "Currently listening to {song title}" has been a part of ambient status displays in IM interfaces since back early in the ICQ days.
I've known people in office settings automate their IM status (back to at least Lync/Skype for Business before Teams) to say things exactly like "making a presentation" or "busy coding", whether for fun or for micro-managing bosses or nagging coworkers hoping to digitally "peak over their shoulder".
Making it opt-out versus opt-in is maybe "gaming context-related" as Xbox and PlayStation both have opt-out versions on their own social networks. But the existence of detailed and automated status messages certainly isn't new from "gaming contexts".
Not to mention that you can obviously disable this.