Copyright isn't something you "get." It's something that comes with the production, no extra steps required.
You've accidentally come upon the other reason why your example doesn't hold in the training data case. How do you suss out who actually holds the copyright on that data.
Take my shitpost for example. Whonis the copyright holder? Me the author? Reddit the platform I published on? What if the source was an Internet Archive snapshot?
Do you earnestly believe this could work in practice?