The real blind spot is that entropy is meaningless for a specific sequence, you can't really ask about the entropy of pi if you don't have a theory for how the numbers are generated. Sure if it is pick a uniformly random real number between 0 and 10 then both files have equivalent entropy, but sending pi is also vanishingly unlikely.
There's actually a more subtle way in which this is a blind spot, which takes a bit more machinery. You can define entropy for an ergodic system, which could be considered a kind of mathematical RNG. Now as it turns out this provides a way to generate something almost equivalent to a particular distribution except that this argument only holds for most starting points not all. A direct example would be how pi generates a perfectly fine random distribution of digits (we think) but something like 1/3 does not.