The random process a frequentist should repeat is flipping arandom biased coin, and getting a random bias b and either heads or tails. You are assuming it is flipping the same* biased coin with fixed bias B, and getting heads or tails.
The probability arandom biased coins lands heads is 1⁄2, from either point of view. And for nshepperd, the point is that a Frequentist doesn’t need to know what the bias is. As long as we can’t assume it is different for b1 and 1-b1, when you integrate over the unknown distribution (yes, you can do that in this case) the answer is 1⁄2.
The random process a frequentist should repeat is flipping a random biased coin, and getting a random bias b and either heads or tails. You are assuming it is flipping the same* biased coin with fixed bias B, and getting heads or tails.
The probability a random biased coins lands heads is 1⁄2, from either point of view. And for nshepperd, the point is that a Frequentist doesn’t need to know what the bias is. As long as we can’t assume it is different for b1 and 1-b1, when you integrate over the unknown distribution (yes, you can do that in this case) the answer is 1⁄2.