@Chris, thanks for the response...as it happens, I've already worked it out with some help from a friend. I'll post my findings here for posterity (in case anybody else was similarly confused about how ELB works).
This would be more clear with a diagram. But the summary is that in each availability zone, you need to create both a public and a private subnet. When you add availability zones to your ELB, you need to select the public subnet for the zone. This had already been done in us-east-1b before I got to this setup, and I had simply missed this nuance of ELB configuration. So for the new availability zone, I had to do this...
us-east-1c private subnet 10.1.3.0/24 (using nat instance as default route) public subnet 10.1.4.0/24 (using internet gateway as default route)
Then my instance goes in the private subnet as expected. And the lynch pin of this whole thing is (drum roll....)
When I add us-east-1c to my ELB, I have to select the public subnet...10.1.4.0. Otherwise the instances will pass the health check (since the ELB can communicate with any instance within my entire VPC) but the responses from the servers cannot make it back out to the public internet.
This is what is so confusing. And I still don't fully understand it. The instance can make a request for, say, www.google.com. I can RDP to it and open a browser and get the web page. But a request from a host (like my laptop at my house) will die. strange.
PS: another note...make sure you are using enough NAT instance for your load. I think we ran into an issue where our NAT instance simply ran out of ports because too many web servers were trying to route outbound connections to 3rd party APIs through it. Quite honestly, I'm not good enough at this level of network/OS troubleshooting to be sure. But my theory is that our 8 instances of IIS were holding too many connections open to the NAT instance. We were also abusing the NIC on that micro instance. I upped us to two large instances, one per AZ and things smoothed back out. Both NAT instances are humming and we're not seeing the hung processes in IIS anymore.