> But that is not what the Church-Turing thesis says. A Turing machine cannot mimic a truly random physical process, pretty much by definition.

It's not clear that a truly random process exists. There exist deterministic interpretations of QM, for instance. Certainly unpredictable processes exist, but that's an entirely different classification (whether a Turing machine halts is also unpredictable, but still deterministic).

Further, Turing machines can generate pseudo-random outputs that pass all known randomness tests.