It does. It's up to engineering to make errors uncorrelated. The google paper being referenced actually makes an "error budget" to see what the main sources of errors are, and also run tests to find sources of correlated errors.

The claim about this is that correlated errors will lead to an "error floor", a certain size of error correction past which exponential reduction in errors no longer applies, due to a certain frequency of correlated errors. See figure 3a of the arxiv version of the paper: https://arxiv.org/abs/2408.13687