Easily reversible hash functions as a method of data transmission
The hashing function you describe would be a magical compression method. Well, I'm sorry to burst this on you, but magic isn't real. Magical compression that can take arbitrary data and condense it to a very small size does not exist. In fact, it's easy to prove mathematically that there is no compression method that reduces the size of all data, by the cardinality argument you give.

Compression that reduces the size of common data certainly does exist. But it works by recognizing certain kinds of patterns. Cryptography doesn't help with recognizing patterns — if anything, cryptography is all about hiding patterns.

Compression that can take data among a finite set and condense it to a fixed size is called naming or indexing. Cryptography doesn't help with that either.