If you assume AES is a pseudo-random permutation(which is pretty much necessary for it to be secure), then yes. However, it's way easier to just use a random IV/noce and keep the keys. Your new key must be distributed to both parties secretly, the IV/nonce can be public, so you can just send it (HMACed of course) in the clear. You obviously can't do this with a key and you may introduce problems depending on how you try to send it.
Typically the IV in CTR mode is called a nonce (shot for number used once). The reason for this is that it, unlike most uses of an IV, need not be random, just only used once. Since we assume the block cipher is a pseudo-random function, we just need to ensure it doesn't get the same value feed into it twice and it will giving us random looking output. Strictly speaking, we just care that the same (Nonce,counter) pair is not used twice, but since we have no control over the counter value, we make sure the nonce is unique.
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Assuming that the block cipher is a pseudo random permutation also tells us we only care if the same (key,nonce, counter) is used. Effectively, the key is just selecting at random the permutation we use from the set of all permutations of a given length( the cipher's block size). For each permutation, the output on (nonce,counter) is unique, so we don't care about repeats under different keys.
