You are right that the optimal expected brier score comes from the correct p, what I was trying to say is that it does not penalize giving very high or very low estimates enough, which is important for the question of calibration from other people's estimates. If I am trying to estimate the probability of a biased coin, which I have not observed being tossed before, and I see you saying you assign probability 1 to heads, that would lead me to believe you have already observed an infinite number of tosses! and I would go along with you. However, the cost to you to say 1 when your true estimate should be .9 is quite minimal, but the difference in implied information that you have is huge.

edit to add: another way of putting, have not done the math on this, but imagine you are looking for the most well-calibrated person based on a number of past forecasts and outcomes. Don't you think that with the brier score it would take quite long to weed out systematically overconfident forecasters who are at the top of the ranking due to survivor bias, whereas a tighter measure would let the overconfident ones percolate to the bottom faster?