The other two downsides are: Some people may chose not to have their email account on the phone. Personally I don't want to carry around access to my main email at all times (the same goes for access to my main bank account, BTW.)
Also, email delivery sometimes takes a very long time, it can be minutes, if you rely on email forwarding to protect your main email address.
Also, email delivery sometimes takes a very long time, it can be minutes, if you rely on email forwarding to protect your main email address.
Auth apps are better for 2FA, at least for me.