If I can trick a user into submitting their username and password on my site, I can send a request to Google using that username/password and trigger a message to the mobile application. The user continues with the flow, enters the code on my screen, and I now have access to their account, same as before.
Yes. 2-factor authentication does not address site impersonation.
This is for the more common case of trying to access a user's account without the user's direct involvement. (e.g., if I grabbed lists of common passwords, try to use your password from a cracked site to access an account on google, etc.).
Classify this as "step forward" not "silver bullet".
Except they specifically list anti-phishing as one of the use-cases:
"if you reuse the same password on multiple sites and one of those sites gets hacked, or your password is conned out of you directly through a phishing scam, it can be used to access some of your most closely-held information."
You're right that it helps in cases where all the attacker has is your username and password. However, the blog post overstates the merits of this feature a little bit. ;)
How is this any different than the prior situation?
Now instead of needing to trick a user into giving you their username and password, you need to trick them into giving you their username, password, and one-time token.
It's designed to address situations where the password is discovered by other parties.
If I can trick a user into submitting their username and password on my site, I can send a request to Google using that username/password and trigger a message to the mobile application. The user continues with the flow, enters the code on my screen, and I now have access to their account, same as before.