I'd say it can be a problem because there have been examples of getting AIs to spit out entire copyrighted passages.

Examples that have turned out to either be a result of great effort to force the output to be a copy, a result of poor training techniques that result in overfitting, or both combined.

If this is really such a straightforward case of copyright violation, surely there are court cases where it's been ruled to be so? People keep arguing legality without ever referencing case law, just news articles.

Furthermore, some works can have additional restrictions on their use. I couldn't for example train an AI on Linux source code, have it spit out the exact source code, then slap my own proprietary commercial license on it to bypass GPL.

That's literally still just copyright. There's no "additional restrictions" at play here.