Also, intuitively I want to be able to use anthropic reasoning to say "there is only a tiny chance that the universe would have condition X, but I'm not surprised by X because without X observers such as us won't exist"

Hmm, that's an interesting angle on the issue, I didn't quite realize that was the motivation here.

I would be surprised by our existence if that was the case, and not further surprised by observation of X (because I already observed X by the way of perceiving my existence).

Let's say I remember that there was an strange, surprising sign painted on the wall, and I go by the wall, and I see that sign, and I am surprised that there's that sign on the wall at all, but I am not surprised that I am seeing it (because I can perform an operation in my head that implies existence of the sign - my memory tells me I seen it before). Same with the existence, I am surprised we exist at all but I am not surprised when I observe something necessary for my existence because I could've derived it from prior observations.