They'll still want your phone number and carrying a second phone just for those moments is a bit of a hassle.

I'd be interested in a dual sim phone that had an instant "private mode" user account switch to the second sim. But even that would be a pita and it would severely limit the choice of hardware as well.

Why do you need to have the phone, beyond the initial registration?

If you just register an account and use it forever, how is that different from "having an active Facebook account"?

Initial registration would happen every time you want to do anything. (And you'd need a new phone... every time...)