There's no way I would give away my phone number to any random website. Thus, I would never use two-factor authorisation that involves my phone, unless it's a bank or something. Most of "social" accounts aren't all that important anyway.
2FA doesn't mean phone number. There's also HOTP and TOTP, dongles, bio, etc.
Most of my two-factor auth involves a time-based one-time pad (TOTP) that involves scanning a QR code once to sync the phone to the service. The service doesn't know what phone or other device I'm using.
You don't have to use your own phone number. I use a Google Voice number for similar purposes. It also works online so I don't need to worry when my phone is out of battery.