I don't think this is all that unexpected given work of Lemke Oliver and Soundararajan 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.

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.

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.

I don't think this is all that unexpected given work of Lemke Oliver and Soundararajan 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.

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.

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. (EDIT: it seems like the bulk of this $d$ term is due to the definitions of $B_0$ and $B_1$; see Will Sawin's answer.)

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.