I don't think this is all that unexpected given [work of Lemke Oliver and Soundararajan][1] on residues of consecutive primes.  (I realize that the first word of that paper's title is "Unexpected", but I don't care.)  Note that John Omielan's first link in the comments points (eventually) to this paper.

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Let's look at the run statistics of the first 49,981 bits of $B_0$, corresponding to about half of the first 100,000 primes.  There are runs of equal consecutive bits of lengths 1 through 7, and the counts of these runs are, respectively, $$(27261, 7710, 1771, 390, 67, 13, 2).$$

Now consider a Markov process with states {0,1} and transition matrix \begin{pmatrix}\frac{1}{2}-d & \frac{1}{2}+d \\ \frac{1}{2}+d & \frac{1}{2}-d\end{pmatrix} for some parameter $d \in [-1/2, 1/2]$.  We should expect shorter runs for larger $d$, and longer runs for smaller $d$.  If we take $d = 0.28$, we get the following counts for runs of length 1 through 11, averaged over 100 samples: $$(30378.2, 6700.82, 1472.6, 324.16, 71.42, 16.01, 3.7, 0.75, 0.15, 0.02, 0.01).$$  This looks like a decent fit to the observed values, if perhaps a bit too low on medium-length runs and too high elsewhere.

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I'm not sure what value of $d$ is realistic here: Lemke Oliver and Soundararajan have (conjecturally) asymptotically correct expressions, but they're studying a slightly different problem.  More specifically, if we look at mod 4 residues of the first $10^6$ primes, their data suggest $d \approx 0.07$ is a reasonable guess for their problem.  For smaller primes $d$ should be larger, but your $B_0$/$B_1$ setup is artificial enough so as to make their methods significantly harder to apply.

More broadly, the shape mismatch in the data suggests (unsurprisingly) that there are longer-range effects going on than those captured by the Markov process modelling.  I expect spending some quality time with Lemke Oliver and Soundararajan and thinking about the prime tuples conjecture should give you better explanations here.




  [1]: https://arxiv.org/abs/1603.03720