You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them.

It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.

Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end").

It's another of those intuition challenging moments in math.

It's the wackiest result I know of, and seems overwhelmingly more profound than BT.