Is it possible to mathematically extract an AES key from black-box encrypt/decrypt hardware?
What you describe is Chosen-Plaintext Attack (CPA) and AES and secure block ciphers are designed to be secure against this. 

Having $2^{16}$ chosen-plaintext under one key doesn't help you to extract the AES key. You have to go to the full-brute force to find the key.

Since you have one target, you cannot get help from attacking many keys simultaneously. In some cases, the black box may reside in many days in front of you, thus, during those days, you will get many target keys.

For $t$ targets, the expected cost breaking one of the $t$ keys is $2^{128}/t$ and that will be far below $2^{128}$. If you have a billion target (~$2^{30}$) the cost will be ~$2^{98}$ to find one of the target keys.