AES-NI itself doesn't provide an avenue for key entropy reduction, since it doesn't generate keys itself, or for exfiltration of stolen keys through the encrypted output, or for introducing mathematical weaknesses into the diffusion properties of the algorithm. If an AES implementation produces output that differs by even one bit from a correct AES implementation, then decryption will fail.
Non-constant timing would also be detectable, though as you say cache side channels are feasible. Power-side-channel key exfiltration is certainly feasible if the attacker can measure power consumption, but AES-NI isn't relevant to many threat models that permit power side channels; amd64 CPUs aren't used in smartcards.
But certainly the IME could steal AES or other cryptographic keys from memory, store them in its own storage, and leak them through some other channel.
Non-constant timing would also be detectable, though as you say cache side channels are feasible. Power-side-channel key exfiltration is certainly feasible if the attacker can measure power consumption, but AES-NI isn't relevant to many threat models that permit power side channels; amd64 CPUs aren't used in smartcards.
But certainly the IME could steal AES or other cryptographic keys from memory, store them in its own storage, and leak them through some other channel.