Looking at the design for Feistel ciphers, they use a list of round keys which are generated from the main key using the key schedule of the associated block cipher. Some block ciphers need this as to prevent repetition, but why does a Feistel network need it?
If $F$ is a good PRF, then the output should be indistinguishable from random after the first few rounds. Under the random oracle model, one would expect that $L_0$ is made pseudorandom after the first round, then $R_0$ is made pseudorandom right after that. Continuing this should never return you back to the plaintext in a reasonable amount of time.
So my question is why are round keys used in the Feistel network as opposed to making all the round keys the same? Did I get anything wrong in my reasoning?