That's basically equivalent to saying that the natural numbers encode all information [0], it's just that the index to say which natural number you mean is rather long.
However for natural numbers, the 'decompression' is very fast. Much faster than for Pi.
That's right. But it does give a feel for what infinity is to our finite minds. It ain't just big numbers. It's all computation! (For a certain definition of computation)
Enumerating the natural numbers is only a single trivial computation. Enumerating the digits of Pi is a bit more involved, but it ain't universal, either.
However for natural numbers, the 'decompression' is very fast. Much faster than for Pi.
[0] You can make this precise via Bijective Numeration https://en.wikipedia.org/wiki/Bijective_numeration to handle leading zeroes nicely.