I think there's some confusion about messaging. If Google knows the model/name, then it's likely not SMS being sent, but rather one of the "review this login" system pop-ups.

See if you can revert to SMS verification through some option at login. I can't verify it now, but the choice could be accessible though something non-obvious like "I'm having problems verifying". Make sure it's trying to use a text message instead of some other method... and let's hope you have it configured.