It all comes down to actual specific statistical tests, and how hard they are to break in specific applications.

No CSPRNG is absolutely perfect, no CSPRNG has ever absolutely passed every statistical test thrown at it.

In MCMC, it stresses very different statistical tests than the typical CSPRNG tests.

Every PRNG is absolutely broken if you want to be absolute about it. MCMC and crypto applications push on different aspects where statistical issues will cause application level failures.

See e.g. this paper https://www.cs.hmc.edu/tr/hmc-cs-2014-0905.pdf

(it's not the end all be all, but it's a good survey of why this stuff matters and why it's different)