Well, pi is almost certainly normal. That means that it stores all possible information (for example, the entire contents of the internet). We haven't proved normality of pi, but if we do, then seeing the face of God in the digits of pi is just the infinite monkey theorem. But this is mundane. You can also see them in the square root of two or in e.

I get that Sagan was trying to get the "numinous", but the tiniest bit of mathematical knowledge makes pi mundane, not mystical. If you want to make me mystical with mathematics, oh boy, talk to me about the Gauss Bonnet theorem, the classification of finite simple groups, or the moonshine conjecture. But pi is pedestrian.

Conversely he did try to cover this with the statistics - the message was found much sooner in the sequence then the statistics would dictate were it random.

But we know this is false. We have computed way more pi than he knew about when he wrote that.

Conversely, we also know that even with the amount of pi we have, we can already come up with any kind of pattern you could want to, by changing the base, by arranging the numbers in a different order, or by picking particular colours or whatever. In a sense, this is an application of Ramsey's theorem. We can find patterns in pi for the same reason that we can see patterns in clouds, or the Bible, or in constellations.

I still think it's cheap potheaded thinking that dilutes the whole "mystery" and "numinous" edge he was going for.