The self-sampling assumption, one of the two main schools of anthropics, claims that: "All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class." It appears to me that, if this were true, then, in a variant of the Sleeping Beauty problem that I shall describe, the participant would have a supernatural ability to predict the future. I am unfamiliar with this area so I apologise for any mistakes I am likely to have made.
Suppose that on Monday, unaware of whether it is Monday or Tuesday, you will enter a laboratory. You will be informed by the experimenter that it is Monday and then watch a fair coin flip. If the coin lands on heads, you will be allowed to leave. However, if it lands on tails, you will be put to sleep with an amnesia-inducing drug that will cause you to forget your experience of Monday's experiment. On Tuesday, you will then enter the laboratory again, still not knowing whether it is Monday or Tuesday, be informed that it is Tuesday and be allowed to leave.
Now, the world in which the coin lands on heads will exist with probability 1/2, as will the world in which the coin lands on tails. Therefore, according to the self-sampling assumption, when you walk into the laboratory, the probability that it's Monday and the coin is going to land on heads is 1/2, the probability that it's Monday and the coin is going to land on tails is 1/4 and the probability that it's Tuesday is 1/4. If the experimenter then informs you that it's Monday, Bayes' theorem tells us that the probability the coin will land on heads is 2/3. You are now apparently in a position to correctly predict a fair coin flip with a probability of 2/3.
I really like this thought experiment, thank you for sharing!
Personally, I agree with you, and I think the answer to your headline question is: yes! Your reasoning makes sense to me anyway. (At least if we don't combine the Self-Sampling Assumption with another assumption like the Self-Indication Assumption as well).
I think that your example is essentially equivalent to the Doomsday argument, or the Adam+Eve paradox, see here: https://anthropic-principle.com/preprints/cau/paradoxes But I like that your thought experiment really isolates the key problem and puts precise numbers on it!