I think copyright is mostly not relevant. Contracts operate mostly independent of copyright law (in the U.S.). If OpenAI puts in their license something to the effect of "you may not train your LLMs on these outputs" and/or "your access is limited in these ways", but you violate the license, they can and should block your access and sue you. These are things that can be monitored and enforced under existing law.
And, in fact, they already do this:
- OpenAI [1]: "[you may not] Use Output to develop models that compete with OpenAI."
- Google [2]: "You may not use the Services to develop machine learning models or related technology."
- Anthropic [3]: "[You may not use our services] to develop any products or services that compete with our Services, including to develop or train any artificial intelligence or machine learning algorithms or models or resell the Services"
[1] https://openai.com/policies/row-terms-of-use/
[2] https://policies.google.com/terms/generative-ai/archive/2023...
[3] https://www.anthropic.com/legal/consumer-terms