Discord often solves this by showing an alert when you login. Product updates and other news often shows up there. You have to close it to get into the app.
The mindset around account recovery would be: if you value your information in your account, add an email. Otherwise, you can start again if you lose it. Doesn't work in all situations.
Seems like a great idea... until a few years down the line you run into an issue for which you need to send a message to your users. And you can't.
These are often legal issues. Needing to forward on take-down notices, changing privacy laws (remember all of the emails that were sent out when GDPR came into effect).
Legal issues aside, the killer challenge is account recovery. People forget their passwords. How are you going to give them back access to their account without a verified communication mechanism?