Why do Feistel ciphers need round keys? 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?
Iterated ciphers need variability between rounds to resist so callad Slide attacks. One common way to thwart this attack is with a key schedule generating different round keys for each round.

Slide attacks exploit the repeating rounds of the cipher by finding a collision between one input plaintext and the intermediate value after one round of encryption of some other plaintext. This collision gives a "slid pair" and gives the attacker two known input/output pairs of the round function which in many cases is enough to recover at least parts of the key.

In the question we have a balanced Feistel cipher where every round uses the same round function and key. Due to the birthday paradox we expect to find a slid pair with on the order of $2^{n/4}$ chosen plaintexts or $2^{n/2}$ known plaintexts, breaking the cipher.