Did you see the bolded part about how once there is any amount of matter in the nearby spacetime, the mathematical solution that yields white holes no longer works? My understanding is that the BB singularity, on the other hand, is under no such restriction. I believe this is one of the many differences that separates the BB singularity and white holes.
tommac, you appear to have great curiosity and a good grasp of certain basic concepts. But you seem to have rather significant gaps to fill as well, and speculating and asking questions on internet fora are no substitute for reading and studying on your own. I suggest starting a thread (or two) in Q&A to solicit recommendations for books on topics that interest you the most.
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