Specifically, if you set things up as in the linked article, no traffic or other information is going to Google (unless you think that the Google Authenticator app is leaking info to Google for whatever reason). Specifically, Google doesn't see how many logins your app gets.

If you're trying to sneak up on a market, or if for some reason you're trying to hide from Google the number of active users you have, you might see this as an advantage.

I might consider it an advantage if Google isn't able to snuff out my user's accounts/access with impunity. I don't understand the technology enough to know whether that is the case, but I wouldn't rely on Google for authentication at this point for this reason.

TOTP is surprisingly easy to understand and implement. I do my best to explain how it works in the "Understanding TOTP" section of this article. Take a look at that section and let me know if it makes sense.

What's ambiguous about it is whether or not I need a Google account (and whether Google having taken away that account, I can still log into your service). The "Understanding TOTP" section seems to be saying that you don't and they can't, in which case there's nothing to object to.

Great feedback, thanks. I've updated the "Adding Google Authenticator" section with a note that should make that less ambiguous.

That's a valid point, if you don't rely on Webmaster tools and analytics. If you do, they can extract this information anyway.