I don't disagree with anything you said but the law as it stands today favors the copyright holder and whether we agree or like it or not in this case (It goes both ways too) Twitter's policy of removing first and having the ability for someone to contest a claim after the fact is their best option of staying in the clear.

I don't side with Twitter on much, if anything but I can't fault them for removing that image on liability grounds even if he is president and its fair use etc. Perhaps his team is contesting it and they'll be forced to restore it, in this case it is possible it is not political.