Why do Feistel ciphers need round keys?
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.