I'm not sure I can agree with this analysis.
Consider two hypothetical universes. In one, p = 10^-1000, but by a statistical miracle, here we are. We would certainly be alone in this universe, because the number of stars in the visible universe is inconsequential compared to this number.
In the second one, the universe is the same size, but p is such that there are 100 planets with intelligent life like ours.
If you cannot say "anything at all about P" from our existence, then we should be indifferent between these two hypotheses, on the basis of the evidence of our existence.
However, this is not the case. The evidence supports the second hypothesis much better than the first. If X occurred, and is more likely under hypothesis B than A, then X is evidence for hypothesis B, otherwise you don't have a consistent system of probability.
Now you can further imagine that God comes down from heaven and tells you that in fact both of these two universes actually exist, and they are the only two universes that exist, so you are definitely in one of the two, but doesn't tell you which one. Is your posterior probability now 50% between the two?
"God" here represents the anthropic principle leaning over and saying "psst, don't worry about the 10^-1000, because, well, here you are". But you still have to worry about the difference between 1 and 100. If N beings in universe A and 100 * N beings in universe B all face this same question, your only reasonable conclusion as one of them is that the odds are 100-to-1 in favor of you being in universe B.
So it's not the case that our existence tells us nothing about p. Of course, p is what it is, and only physics and chemistry and biology can actually reveal it, but the fact that we are here means that theories that make our existence much less likely are themselves much less likely.