Why do block ciphers use key schedules instead of round constants? (Even-Mansour) Let's take AES as an example. What would be wrong with just having a 256 bit key that you XOR into your input and then XOR into your output? No key expansion at all.

I believe it's even known as the Even-Mansour construction and is secure under the assumption that the permutation is sound.

So I'm confused by the purpose of key schedules, you need variability to resist slide attacks but you don't need a key schedule for that, round constants suffice.
What would be wrong with just having a 256 bit key that you XOR into
  your input and then XOR into your output?


If I understand you properly, you split the 256 bit key into two 128 bit parts. This will be the whitening key that helps to resist exhaustive key search.

In term of round constant, it is adds asymmetry to the key and prevents slide attacks. However, you overlooked other attacks such as invariant subspace attacks for which you have to choose constants that eliminate weak key classes. 

A related-key attack may remove the effect of the constant (key difference  eliminates the constant) but in real world, the adversary needs to have access to key generation to perform this attack. 

I believe that the design of key-schedule lacks to solid proof. There was a paper in FSE18 about the mathematical proof of the AES-128 key schedule under related key attack. A solution was proposed using a better and lighter key-schedule without s-box or round constants, just permutation.

However, how to choose constants that prevent slide attacks? In other words, is there a proof that adding any constant will stop a slide attack?