This is incorrect.
You only need the ability to generate TOTP or U2F tokens. This is often done using a smartphone app, but can also be done by a desktop app like 1Password or a hardware device like a Yubikey: https://github.com/blog/2071-github-supports-universal-2nd-f...
It's things like that that make me wonder why TOTP tokens are supposed to be conceptually different from passwords. A TOTP scheme involves knowing a master password, and nothing else.
Why? You're not any less secure by using a personal phone. What are the odds that an employee is going to be phished and have their phone compromised by the same entity.
I'm already answering emails out of office hours which is for my employers benefit and they want to functionaly own my phone because of it?
For companies that don't do that Github also offers the option of FIDO U2F compatible keys.
Do you give every enployee a mobile phone, or do you ask your employees to use their own personal phones?
Asking them to use their personal phones seems like a very bad solution. Many software companies do not routinely give developers mobile phones...