Using classic Outlook for Windows in business environments
And that is the reason why you supply MANY different methods for the second factor. I use 2-Step verification on my Gmail account and, beyond the fact that my cellphone is an Android and, as you point out, the first thing Google does is light up the phone, I also have it set to text two different cellphones and voice call a landline. Next, they allow Authenticator apps. Google offers its own, I personally prefer the Microsoft Authenticator. Google will also send you a list of 10 single-use codes, if you wish. You should set up as many of these and you can so an issue with any single method won't bring the whole world to a stop.
I do not know why your phone notifications aren't showing up, that's a Google support issue. Yes, if you disable 2-step verification, the App Password will no longer work, and you'll lose access form Outlook.