I think it is better to say that bet on offer is fair. It is not a property of just the coin, but also of the bet. We do not notice that there is a choice of bet because it is even odds (which corresponds to max ent), but for any weighted coin there is a corresponding fair bet.

Fair bets do have lots of special properties, but we would have the same situation if a correct choice of tails paid 1 and a correct choice of heads paid 2: Alice and Bob would both always bet H. (except in the 1/1000 chance that we start with 10 Ts and Alice updates wrongly; but the asymptotics are the same)