At this point, 2FA is the only thing I use my phone for anymore. It's the only reason I even have a phone; I spent about a year without one until I had to for 2FA. But I don't need to carry it around anywhere for that. It would be inaccurate to call it a "mobile" device.
What about extending the protocol to an actual channel-bound challenge-response one, without the need for a (risky) out-of-band key exchange via a QR code?
We could call it something like Web Authentication. I could even imagine small, keychain-sized USB authenticators that you have to touch a capacitive button on to approve an authentication :)