Okay, I was wrong. I just did a big update to the circuitry around the MAX4468.
That had been bothering me a bit, and I wanted to see if what I had could work with a piezoelectric mike, so I did a little digging and found that my approach was (as far as I could figure) altogether wrong, and would probably not work with an electret either. The electret was the main point (the piezo is only secondary), so I needed to fix things.
The recommendation I found was basically to rely on the schematic for the MAX4466. I'd switch to that chip altogether, but now that I have a grasp on what the Shutdown does I'd like to keep that functionality. This does leave the Bias pin open, though; I'm not sure whether I should do that (which is my impression at present) or hook it up to Ground, 3.3V, or IN+.
I also moved all of the electronics for the MAX4468 as far west as I could (though there's probably room for even more), just to leave more room for new electronics for the NCS2211 (or something altogether new).
Anyway, I've updated the parts list, and I've also kept the previous version of the board here in case anyone wants to compare them (and tell me that I had the whole thing right to begin with -- even though I'm pretty sure I didn't).
I still wouldn't want to try this with a piezoelectric mike without breadboarding first, but I think I at least have a model that would be worth trying out on an electret now.