OK, I figured out a messy workaround. According to this support page, Twitter API no longer supports RSS feeds AT ALL. There are third party services that will turn your Twitter into an RSS feed.
In /extensions/twitter/extension.inc.php I altered this function (at line 61):
function get_url()
{
return 'https://www.twitter-rss.com/user_timeline.php?screen_name='.$this->get_option('username');
}