After a quick look at their ToS I too do not see a clear mention.

In a previous HN discussion from September 2021:

[zorkian] It's still against the TOS -- my point is only that we don't specifically look for third party client users and we have no specific plans to do so.

[Note zorkian’s HN profile: “VP, Core Tech @ Discord. Opinions are my own and I do not speak for my employer (unless noted explicitly.)”]

[tumult] Just curious -- where in the TOS are third party clients forbidden? I've read it a few times, and I'm not seeing it.

[dge2020] Probably:

You agree not to (and not to attempt to) (i) use the Service for any use or purpose other than as expressly permitted by these Terms;(ii) copy, adapt, modify, prepare derivative works based upon, distribute, license, sell, transfer, publicly display, publicly perform, transmit, stream, broadcast, attempt to discover any source code, reverse engineer, decompile, disassemble, or otherwise exploit the Service or any portion of the Service, except as expressly permitted in these Terms;

It's a catch-all but the best restriction that prevents third-party clients is probably 'prepare derivative works based on' and 'reverse engineer' (which you would need if you want your third party client to use any regular client API calls, or if you want to support signing in with the user-facing login page/qr login).

[tumult] A third party client like Ripcord isn't a derivative work. It also wasn't made by reverse engineering any of Discord's code. Even if it were, these rules still wouldn't apply to using third party clients.

So it seems that, as you note, the ToS do not explicitly mention third-party clients but Discord themselves believe that it forbids third-party clients.

https://news.ycombinator.com/item?id=28435490