> By the time you add all the whitening and that sort of thing to make those problems go away I'm not sure how the information theory proofs hold up.

Why wouldn't the information theory proofs hold up?

Fun puzzle: Suppose I give you coin you can flip with some bias (weight). Maybe 75% of coin flips are heads. Maybe its 65%. You don't know.

You want to generate a sequence of bits with uniform randomness (exactly 50%). Without first sampling the coin & calculating the bias, how do you generate the uniform sequence of bits?

The answer to that puzzle would probably work fine for your theoretical one time pad.

Also, in practice I suspect most of the obvious ways your random noise circuit could be broken could be detected using statistical methods. You can't use math to prove a random number generator is truly random. But you can certainly detect a lot of common failure modes of "random" number sources.