Why doesn't Twitter have 2FA or U2F? Problem solved, at least in terms of users not losing reputation within their own social space because someone is posting as them.
They do. In fact, I just logged in to change my password and had to go get my phone from another room so I could read the code. The annoying thing is that they don't support Google Authenticator or some other TOTP/HOTP token generator. It sucks trying to log into services that use SMS based 2FA when you are not in an are where you get cell reception.
App based 2FA seems extremely limiting. I don't want to install their app on my phone.
Doesn't have to be just their app. There are plenty of apps and chrome extensions that mimic authenticator ability.