What are the disadvantages of XOR key derivation? Assume a 256-bit Key and a 256-bit Salt, with uniformly distributed entropy, both fully independent.

Assume Key is secret and Salt is public.

What theoretical advantages would deriving an AES-256-CTR encryption key using Key XOR Salt give to an attacker?

Assume that the product of Key XOR Salt will remain secret.

Assume that a KDF such as HKDF or a PRF such as HMAC cannot be used.
The two main concerns I'd have with the scheme you describe are 1) the risk of one of the subkeys being compromised, and 2) related-key attacks on the cipher.

The first concern is obvious: since the XOR operation is reversible, if an attacker finds out any one of the subkeys and knows the corresponding salt, then they can recover the master key.

Of course, if you can guarantee that all the subkeys are just as well protected as the master key, then this may be a non-issue.  But an extra layer of safety in case of subkey compromise can still be a nice thing to have, and using a non-reversible key derivation function instead of just XOR provides it at a very low cost.

The second concern is a bit harder to quantify.  There are known related-key attacks against AES, but so far none of them are efficient enough to be a real threat in practice.  Still, they do demonstrate that AES does not necessarily deliver its full advertised level of security in situations where an attacker may know the XOR of two or more keys.

As before, assuming that no new and significantly better related-key attacks against AES are found, you might be able to get away with this in practice.  (We do need to make some assumption like that anyway, given that nobody has proven AES to be secure, with or without related keys.)  But again, the safe and conservative choice would be to use a proper key derivation function that does not reveal any mathematical relationship between the subkeys.