Generally, I'm always wary of "out of sight, out of mind" type solutions - for any kind of problem. In this particular case, this assumes harrassers only want the original author to know (and even then are interested in the actual outcome of their message).
I think the "shitstorm" dynamics are more complex and there are different motivations for people to take part in one. E.g., it could be simple venting, virtue singalling to peers in your brigade or you sharing the tweet to others in your circle - nothing that needs any perticipation by the original author.
In extreme cases, an author might not realize what is going on until they find outraged articles about themselves in the news.
See the comparable problem of cyberbullying for an example. There, it's often completely irrelevant whether or not a victim knows about the messages that are written about them. Indeed, they can be bullied on online platforms where they don't even have an account. All that is required is for participants in the bullying to know how to access the messages.
Of course, harrassing PMs are a different issue. Here, you can be sure the messages are exclusively directed towards the author and the harm is being done by having the author read them. Blocking those kind of messages would stop the bullying action itself and could work. But then again, this is not so much different from the oldfashioned blocking we already have.