Your example acually exists. It's called actually useful horoscopes. It's creators are from Less Wrong.
That's pretty interesting. I wonder what the authors do with the statistics they're gathering.
One of the features of this program is that the chance of a given horoscope being displayed are affected by how well it has worked in the past
Actually Useful Horoscopes contains very few actual predictions. It's pretty much all generic good advice like "You should be sure to change your passwords now and then" or "Set a good precedent for yourself today."
Regarding the original post, I don't see how reading a useful self-fulfilling prediction involves irrational thought.
You are made of biases. Every time you encourage yourself to think rationally, you are using biases to do it.
For example, the Forer (or Barnum) effect leads people to believe that vague and general descriptions of personality traits are very accurate when they believe that the results are tailored to them (based on some attribute like astrological sign).
If there were a service available to you that would only send you predictions like "you are more likely to notice and compensate for your biases today" would it be worth signing up for or is there a Shelling point around not deliberately engaging in irrational thought? Is there a way to calculate utility for limited self-deception?