I realized the other day that I'd left out a critical portion of the whole process: card registration. You still need the EyeFi-supplied software to register the card for the first time, or to add and remove online services. It might be nice in the future if EyeFi could make registering or adding and removing photo services a web-only procedure that didn't require running the manager software at all. I'm pretty sure the card doesn't actually get consulted when you change the web services, so this should be possible in theory.
I deleted my account, and recreated it. Interestingly, the software didn't to anything to the card when the account was deleted. That seems to indicate to me that all of the smarts about the card registration are on the EyeFi API server. The manager software doesn't do anything like tell the card when it is not registered any more. The card can figure that out on its own from the server.
This is supported by the fact that the very first thing the card does after it gets power or sees a new picture is do a DNS query for api.eye.fi, then POSTs some XML to it with its MAC address and a field called "cnonce" that looks like an MD5. These folks have looked into how this works in more detail.
I wouldn't mind getting my hands on another brand-new card to see exactly what the card's behavior is when it is fresh out of the box. I wasn't smart enough to trace it from the beginning.