Hopefully this will help someone else wrestling with wi-fi on the iPhone:

This wasn’t really a wi-fi problem, but could be mistaken for one. I had no problem connecting and authenticating to my home wi-fi network, but I still couldn’t access the Internet.

Symptoms: can connect and authenticate, get a strong wi-fi signal, can access local machines (by IP address), can access outside machines (by IP address), anything involving a DNS lookup fails

So it turned out that this was a DNS resolution problem. The first DNS server being provided by my provider was unavailable, and the iPhone failed to skip ahead to the next DNS server in its list. I edited the DNS servers and removed the invalid server, and now everything works. (I also confirmed that when the DHCP lease is renewed, these DNS edits remain.)