Facebook recently got told by the CJEU that, no, they can't use people's posts to target advertisements. Even if those ads are what's paying for the platform. That you can't claim such processing as "part of the contract" unless it is absolutely necessary in the same way the post office needs an address to send a parcel.
If Facebook can't even do that, there is no way LLMs will be allowed. (And remember. The GDPR does not care if your system doesn't distribute personal data. Any kind of processing at all falls under the GDPR's requirements)
OpenAI is already being chased by the EU's privacy agencies. Right now they're in the process of asking pointed questions, things will heat up after that.