What they are getting at is that thinking of this in terms of the "quality of the randomness" is thinking about it in quite the wrong way that leads one right u

What they are getting at is that thinking of this in terms of the "quality of the randomness" is thinking about it in quite the wrong way that leads one right up the garden path; so stop even thinking about it like that. Discard that mental model. The randomness has the same quality. It's the same pseudo-random number generation algorithm. Only in one case, the world knows your seed value, and can predict anything that you do that derives from pseudo-randomness; whereas in the other case, the world does not know your seed value, should your seed value be discovered somehow you are regularly re-seeding anyway, and the world cannot predict your actions.