I'm analysing how different components of AES influence its security. However, I cannot find a good explanation to what happens if the MixColumn step would use the identity matrix. Intuition tells me, that then AES won't be secure, but why?
If you eliminate the MixColumn (or equivalently, replace it with an identity Matrix), then the resulting cipher will effectively be 4 independent 32 bit block ciphers. This happens because what happens in one 32 bit row no longer has any propagation to any of the other 3 32 bit rows.
This results in a much weaker cipher; not only would that bring up possible chosen plaintext/ciphertext attacks (where you modify one of the 32 bit blocks, but leave the other 3 the same), you would also have ciphertext only attacks with a birthday bound of circa 256k...