Your problem isn't with 2FA, TOTP or SMS, it's with sharing an account that isn't meant to be shared. When you use a feature in a way it isn't meant to be used,

Your problem isn't with 2FA, TOTP or SMS, it's with sharing an account that isn't meant to be shared. When you use a feature in a way it isn't meant to be used, sometimes you're going to get some sub-par user experiences. Should Blizzard support multiple users of the same account? Perhaps, but that's out of scope here.