Imagine the "sufficiently chaotic process" is simply listing out all the possible data sequences in order, for example 0 1 10 11 100 etc.

Then, "referencing a region" of the state is basically the same as just giving the value. The nth value is just n itself.

So clearly that doesn't give you any advantage. Why would randomizing the data sequences make it any more efficient on average?

Although if you specifically choose the ordering of the sequences so that common sequences are easy to represent and uncommon sequences are hard to represent, then you basically have Huffman coding/arithmetic coding.

Because you can bias the process to be more likely to represent data relevant to your domain.

Note: I'm not in any way a compression expert, so while it may be obvious to you/others, it's a little interesting for me to think of compression in this way (sending parameters to a shared process, instead of sharing the raw data).

This is part of modern encryption. To be more specific, we have blocks and a deterministic cryptographically secure rng, we seed the rng with a public key that is broadcasted, we then encrypt blocks and then combine them together with the number produced by the rng and create the next block of encrypted data.

You may be wondering, why do we even do this? The answer is simple, "the image is encrypted, but everyone can see the penguin". In a less cryptic manner, encrypting blocks simply maps them from one space to another, but is a reversible operation and, as any function does, is deterministic and returns the exact same value every time. The consequence of this determinism, is that identical blocks will always produce the same value and thus, the content is still somewhat visible and in some cases it could leak information. By combining blocks together along with a PRNG, we hide the mapping by requiring that the recipient solves the first block before decrypting further data and we don't end up leaking information.

This is same as giving starting digit in π, right? Not usable for compression, and encryption folks would file it under security-by-obscurity.

This is I think what cryptographic hashes are meant to achieve.