This problem isn't addressing password authentication, its the website knowing who you are and that you are legitimate. Websites that collect things such as phone numbers during account creation don't collect your PII as part of your password procedure. They collect it as a verification that you are an actual being and not a fake account/bot. The ease of being able to go through a forgot password thing is just a positive side effect.

This solution would work amazingly for logging in, there's no argument for that, but it doesn't address the elephant in the room: That the website wants to make sure you are a person/legitimate account and not a fake alias or a bot to scrape info, and when you are the only one providing that information that claim can't be verified.