Guilt is an added cost to making decisions that benefit you at the expense of others. (Ideally, anyways.) It encourages people to cooperate to everyone’s benefit. Suppose we have a PD matrix where the payoffs are: (defect, cooperate) = (3, 0) (defect, defect) = (1, 1) (cooperate, cooperate) = (2, 2) (cooperate, defect) = (0, 3) Normally we say that ‘defect’ is the dominant strategy since regardless of the other person’s decision, your ‘defect’ option payoff is 1 higher than ‘cooperate’.
Now suppose you (both) feel guilty about betrayal to the tune of 2 units: (defect, cooperate) = (1, 0) (cooperate, cooperate) = (2, 2) (defect, defect) = (-1, −1) (cooperate, defect) = (0, 1)
The situation is reversed - ‘cooperate’ is the dominant strategy. Total payoff in this situation is 4. Total payoff in the guiltless case is 2 since both will defect. In the OP $10-button example the total payoff is $-90, so people as a group lose out if anyone pushes the button. Guilt discourages you from pushing the button and society is better for it.
We can tweak the experiment a bit to clarify this. Suppose the coin is flipped before she goes to sleep, but the result is hidden. If she’s interviewed immediately, she has no reason to answer other than 1⁄2 - at this point it’s just “flip a fair coin and estimate P(heads)”. What information does she get the next time she’s asked that would cause her to update her estimate? She’s woken up, yes, but she already knew that would happen before going under and still answered 1⁄2. With no new information she should still guess 1⁄2 when woken up.