Sometimes, people just don't like each other. (I have no beef with Swombat.)
I see no harm in promoting the thoughtful stuff (like this article) and ignoring the angry noise.
The mute feature isn't quite enough for this, since naturally people who follow Shanley will tend to retweet other related topics that have the same tenor. So unfortunately the best thing for my blood pressure is to simply unfollow people who have joined this movement.
Which is all a big shame, because I am a fervent believer in equality (gender or otherwise), and am just as violently opposed to people who discriminate against or insult women just because they're women as Shanley is. However, I am equally hostile to people who discriminate against or insult men just because they're men.
One of the many problems feminism has faced is the lumping in of extremists with more moderates with all sorts of other splinter views; it's to be expected that your work may face the same criticisms.