Does AES-CTR require an IV for any purpose other than distinguishing identical inputs?
Well, no, in your case, a constant IV is not a problem.  With counter mode, the rule is that you cannot reuse the same IV with the same key.  However, it is perfectly fine to use the same IV with different keys, and that's what you're doing.

One minor correction to what you have (that doesn't directly relate to your question): you state that there are problems if "a key is used with the same (or a numerically-close one).  Actually, as long as the IVs are different, "numerically-close" IVs are not an issue.  In fact, it is actually pretty common to select counter mode IVs using a message counter (e.g. the first message gets IV 0, and second one gets IV 1, etc.) 

However, I do have one question: how do you expect someone to decrypt the ciphertext?  If what you're doing is generating a hash of the plaintext, why don't you just go with the SHA-256 hash of the plaintext?