It feels like an awkward middle ground. Like this method has a limited entropy output for the first 2000 coin flips (first 1000 double-heads/double-tails/mixed entries), and then suddenly it adds back a ton of lost entropy.

An other commenter linked to some papers for asymptotically optimal entropy generation, I wonder if there is more of a streaming method there. It feels like there has to be, even maybe after a slow start. My naive intuition is that after 1000000 coin flips you have a good idea what p is, and then you can basically do arithmetic coding from there. Of course a theoretically correct method can't do exactly this, but it might asymptotically approach it.