Ask HN: Mental model for Google account verification?

I have a google account that has two devices associated with it, one Android and one iPhone. 2FA is enabled, and I have an authenticator associated as well.

I would like to find out if I have the mental model of the google account verification process wrong, or it is just horribly engineered.

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First problem: When accessing a Google service on a laptop that has not accessed that service before, Google sends "Is this you" verification to only one phone (the Android phone that belongs to someone else), not to iPhone (mine). I'd like to get notifications on my iPhone as well.

I have tried the following on the iPhone: a. Re-installed google apps (google, gmail) and logged in again, for which I needed to get it approved by the Android phone.

b. Enabled notifications. I did get a notification once on the gmail app, but never after that.

c. Enabled background app refresh

d. Set notifications on Wifi and cellular. Battery levels are fine (apparently, by default notifications are disabled on low battery level)

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Second issue: How does one log in using an authenticator? It never asks me. As soon as I enter the password, it says to tap a Google code on the Android phone. It does not show Authenticator as an option when I click on "Try another way". So frustrated.