When you get to the future, would you regret having to answer "Yes", given that if you weren't so rational you could just answer "No"? If so, you should answer "No" every time. What is the difference between this time and the future? From your present position, you see all future possibilities, and so you make tradeoffs between them, acting on expected utility. But at present, you could similarly take notice of the alternative possibilities that are "sideways" from where you are, things that could happen to you but didn't, and similarly act on expected utility. There doesn't seem to be any fundamental reason for discarding (ceasing to care about) possibilities on the basis of not happening to be located within them, it's just practical to do so since you normally can't do anything about them.

(See Counterfactual Mugging and UDT for more discussion.)

It seems to me that an agent that self-modified to answer "Yes" to this sort of question in the future but said "No" this time

This strategy is not reflectively consistent. From the new TDT PDF:

A decision algorithm is reflectively inconsistent whenever an agent using that algorithm wishes she possessed a different decision algorithm.

If an agent implemented your strategy, they would change decision strategies every time they come across a predictor that flipped heads.