When I initially asked this question, my motivation was a thought experiment: Could a cryptographer from the early 20th century break the "Rube Goldberg" cipher machine in my question? At that time, I knew little about cryptography. Several years later, I've learned a lot about security, and now I can answer my own question.
Data compression, especially when an unknown dictionary is used, appears to be already an effective method of data obfuscation by itself. It, by no means, provides real security by modern standard, but it does make cryptanalysis a lot harder than a classic substitution cipher.
Recently, there was a real-world attack of this nature during the reverse engineering effort of Intel ME firmware, providing a good case study. Intel ME is a coprocessor on modern Intel CPUs. Because it can perform many low-level hardware operations, it was seen as a huge attack vector and attracted many critical examinations by security researchers. CPU initialization and bring-up are the most critical functions executed by Intel ME, however, their corresponding firmware modules, BUP and KERNEL were compressed by Huffman encoding with an unknown dictionary. It made analysis impossible for a while before it was cracked using a known plaintext attack, as described by the researchers in this article.
It was discovered that several modules were have been compressed by LZMA and Huffman simultaneously. The LZMA copies could be decompressed and allowed one to obtain a large collection of known plaintext.
[...] identically named modules in different versions of ME 11 firmware can be compressed with different algorithms. If we take the Module Attributes Extension for identically named modules that have been compressed with LZMA and Huffman encoding, and then extract the SHA256 value for each module version, we find that there is no LZMA/Huffman module pair that has the same hash values. But one should remember that for modules compressed with LZMA, SHA256 is usually computed from compressed data. If we calculate SHA256 for modules after LZMA decompression, a large number of pairs appears. Each of these module pairs yields several pairs of matching pages, both with Huffman encoding and in unpacked form.
After a large set of plaintext-ciphertext pairs were collected, the meaning of 70% of sequences from the code table and 68% from the data table was determined by using linear algebra.
Having a large set of pages in both compressed and uncompressed form (separately for code and for data) allows recovering all of the code sequences used in those pages. The methods needed for this task combine linear algebra and search optimization.
[...] we can set up a system of linear equations: the unknowns are the lengths of encoded values, the coefficients are the number of times a particular code sequence is found in the compressed page, and the constant equals 4096. Code and data pages can both be "plugged in" at the same time, since for identical code sequences, the lengths of encoded values should be the same. Once we have enough pages (and equations), Gaussian elimination provides the one valid solution. And once we have the uncompressed plaintext, length of each value, and their order, we easily derive which sequence codes for which value.
However, the previous method couldn't be used if the length of the codeword is unknown. Fortunately, the SHA-256 checksum of a module allowed one to launch a brute-force attack. Additional heuristics were used to increase its effectiveness.
If we have a sequence of unknown length, another row is added to our system of equations and we can quickly determine its length. But if we don't have the plaintext, how can we determine the value? [...] Fortunately for us, the metadata contains the SHA256 value for the unpacked module. So if we correctly guess all unknown code sequences on all the pages that make up a module, the SHA256 value should match the value from the Module Attributes Extension.
With a computer, it's entirely possible for a good cryptanalyst to crack such a cipher. However, without a computer, it would be extremely difficult for a cryptographer from the early 20th century to crack Huffman-compressed data with an unknown table by hand, even without the use of classical cryptography. But since compressing messages is equally impossible without a computer, my original question was self-contradictory and ill-formed.