>the coin should should converge to 50% of the flips reading Heads

This is incorrect. The convergence is to: (current heads + 0.5(n flips)/

(current flips + n flips).

For a gambler flipping coins, If they are neutral at the time of prediction, you expect them to end neutral.

If they are up flips when you enter the room, you expect them to the run up flips.