That reminds me of the Enigma machines in WWII: The thing was set up such that a given letter in plaintext could not end up as that same letter in cyphertext. T

That reminds me of the Enigma machines in WWII: The thing was set up such that a given letter in plaintext could not end up as that same letter in cyphertext. This was supposedly to add security, but it turned out not only to be a weakness, but one that was critical in breaking the code. That one makes sense, in a way: If you see a S in cyphertext, you know the plaintext did not have an S in that position. That eliminates 4% of all possible plaintexts, and by extension 4% of all private keys. That's kind of a metric ass-ton of entropy you just lost right there, for every single character. My guess is, your proof works by a qualitatively similar principle: Restricting the keyspace, no matter how trivially, gives the attacker information. And encryption is all about magnifying information.