Looking back, I'm not certain if I've answered the question.

I think so: I'm inferring from your comment that the principle of indifference is a rationale for treating a deterministic-but-unknown quantity as a random variable. Which I can't argue with, but it still clashes with my intuition that any casino using the millionth bit of pi as its PRNG should expect to lose a lot of money.

I agree with your point on arbitrary seeding, for whatever it's worth. Selecting an arbitrary bit of pi at random to use as a random bit amounts to a coin flip.