[coppsilgold]

Does there exist a single MCMC example that performs poorer when fed by a CSPRNG (with any fixed seed, including all zeroes; no state reuse within the simulation) as opposed to any other RNG source?

[SAI_Peregrinus]

If there did, it'd be a distinguisher attack on that CSPRNG. So for a non-broken CSPRNG, the answer is "no", by the definition of "non-broken CSPRNG".