Yes.

What if we start with a different, "curvier" prior? After the first toss the probability density should still pivot around the 0.5 point but because it's not a straight line the probability mass in [0.5-x, 0.5+x] will not necessarily remain the same.

Provided the prior is symmetrical, the probability mass in [0.5-x, 0.5+x] will remain the same after the first toss by the argument I sketched above, even though the probability density will not be a straight line. On subsequent tosses, of course, that will no longer be true. If you have flipped more heads than tails, then your probability distribution will be skewed, so flipping heads again will decrease the probability of the coin being fair, while flipping tails will increase the probability of the coin being fair. If you have flipped the same (nonzero) number of heads as tails so far, then your probability distribution will be different than it was when you started, but it will still be symmetrical, so the next flip does not change the probability of the coin being fair.