It might depend on your jurisdiction, but LLM output is usually not copyrightable because it is not made by a human. There might be a few exceptions, if you guided the LLM in very specific and particular ways. But in general, the output of an LLM is public domain the moment it was produced.
Can you never tell anybody and just slap a license on it? Sure.
The issue here is not the license, but that you violate their ToS (if that is valid and enforceable is a different question of course). But if you publish the LLM output on github, and someone else takes it to train their LLM, and you did not actively encourage or help them, it's fine.
Of course it exists in Germany. If I say ‚Hello‘, I could not claim copyright on that sentence because it does not have the required schöpferische Höhe (threshold of originality). The sentence is simply not copyrightable, putting it into public domain.
Note that this is different from Gemeinfreiheit, where a copyrighted work essentially becomes public domain 70 years after the death of its creator.
Can you never tell anybody and just slap a license on it? Sure.
The issue here is not the license, but that you violate their ToS (if that is valid and enforceable is a different question of course). But if you publish the LLM output on github, and someone else takes it to train their LLM, and you did not actively encourage or help them, it's fine.