Counter mode with $\operatorname{AES}_k(m)$ vs $\operatorname{AES}_m(k)$
I understand that it would be quite inefficient as the key schedule would need to be re-calculated for every block, but would it make cryptanalysis easier?


It would make cryptanalysis trivial.

If the attacker knows that

$$P_i \oplus C_i = E_{n \mathbin\| i}(k)$$

and he knows $n, i, C_i$ and has a guess for $P_i$, he can recover $k$, and use that to decrypt everything else.

This works because, with AES, $E_k$ has an inverse $E^{-1}_k$ that's efficiently computable, assuming you know $k$.  There's no corresponding "inverse" corresponding to the message block, and so the "message block" and the "key" inputs to AES are not interchangable from a security standpoint.