The accessability problem of screenshots cannot be understated. Screenshotting and sharing says, first and foremost, that blind and visually impaired users do not matter and you are happy to exclude them. If that’s a message you’re okay with sending, then by all means have at it.
It’s also important to note that screenshots, especially from Twitter and Facebook, have become a means of spreading misinformation and misrepresentation. It is trivially easy to make a fake Twitter post or Facebook post, screenshot it, and spread it around to impugn the purported author or to falsify testimony about events. On the heels of the recent controversy about Melania Trump’s plagiarism of Michelle Obama’s speech, a screenshot of a fake tweet from President Obama praising Michelle’s speech was created using the words Donald Trump had used to praise Melania, to make it look as if Donald Trump had plagiarized him. Thousands of people shared it on every social media platform.
Live links are great. Archive tools like http://archive.is which freeze an URL (like a tweet or Facebook direct link) in time are great. Transcribed and described visual media are great. Screenshots, however, are always suspect and should never be taken at face value by those of us with the privilege of being able to see them at all.