My understanding is Intel has been on 22nm while ASICs have been 28 or 40nm (?), so Intel has inherent advantage if they decide to make a ubiquitously used instruction more efficient. As encryption becomes more used on everyone's computer, Intel is going to have an economic incentive to make AES-NI (and perhaps SHA-2?) more efficient.
You're right in that the AES-NI unit on Intel CPUs will be faster than on any ASIC, but you'll never have thousands of such units on a CPU as you could have on dedicated ASICs.
Thousands of AES units won't do anything for you. This is not a Bitcoin-style PoW using AES (as a hash function) instead of SHA. Most of the Intel chip is being used for mining. In fact one might surmise (in part of course) that the algorithm was designed by looking at the layout of an Intel CPU and working backwards from there.
Ah, that's interesting, I wasn't aware. Thanks for explaining!