If cryptographic hashes are completely unique, could they theoretically be used to transfer data?
No.

They can be used to identify data you've stored elsewhere.  For example, you can probably find a preimage under MD5 for 6f5902ac237024bdd0c176cb93063dc4 by searching the web, and it might even be meaningful English text.  And you probably won't find more than one preimage.

This is the idea of content-addressed storage, which is widely used, e.g. by Tarsnap—although it would be a mistake to use MD5 or SHA-1 for content-addressed storage, because collisions are readily available to anyone who wants them, which have hilarious effects on systems doing just that.

But even if you had a practical preimage attack on MD5—say, an array of quantum computers large enough to run Grover's algorithm—it would give you a preimage, but not necessarily (and probably not) the preimage I have in mind that you'll find with a web search.  Instead, it would probably give you 128 or 192 or so bits of gibberish.  And if you ran it for longer it would probably give you many such blocks of gibberish, long before finding anything that resembles English text.

If the messages you want to transmit cover only a small subset of all bit strings, and are structured with a distribution you know something about, you could invent a better compression scheme for it.  Maybe if there are few enough messages with convenient enough structure, you could compress them into 128 bits.  This depends on specific information about the distribution of messages, and it depends on having no more than $2^{128}$ pigeons to stuff into holes.

But a preimage-resistant cryptographic hash function like MD5 would be about the stupidest compression scheme imaginable, because not only was it designed without knowledge of your distribution of messages, but it goes to great lengths to obscure the original message and to make ‘decompression’ as hard as it can so that the best way is to guess a possible original message and check it.