In theory, a block cipher is broken if an attacker can even tell the difference between application of the cipher and of a random permutation, different for eac

In theory, a block cipher is broken if an attacker can even tell the difference between application of the cipher and of a random permutation, different for each possible key, more efficiently than brute force (i.e. trying every possible key). Since encrypting the same plaintext with a bunch of different random permutations would not help an attacker recover it, I believe an attack like you describe would not be possible without breaking AES.