I mean, the issue the RIAA is raising does not seem to be on AI training, but piracy:

The RIAA has asked Discord to shut down a server called “AI Hub,” alleging that its 145,000 or so members share and distribute copyrighted music: Shakira’s “Whenever, Wherever,” for instance, or Mariah Carey’s “Always Be My Baby.” These songs, and several others by the likes of Ludacris, Stevie Wonder, and Ariana Grande, were named in the RIAA’s June 14 subpoena to Discord (pdf).

The music files were being used as datasets to train AI voice generators, which could then churn out deepfake tracks in the styles of these singers.

Later in the article:

It wasn’t clear, from the RIAA’s letters, whether the body was complaining about the databases of original music or about the AI tracks being generated out of them.

Like, I'm sure they're spooked by AI generated tracks and losing control of the industry... but this seems like a pretty clear cut case of shutting down a Discord server engaged in music piracy.