Agreed. It would actually be a fairly perverse argument to claim that most AI output is somehow NOT owned by the AI provider…

Why wouldn’t they claim ownership of the AI output? They likely already claim ownership of the “transformation” (AI training) of the (pirated) input data.

Exactly. We as users have zero way to confirm they are honoring even the letter of these agreements, much less the intent. And it's super easy for them to weasel around and find a way to cheat while still having a legal claim to honoring the contract. And if you've forgotten, all of these companies are built on a foundation of ignoring copyright law.