In exactly what sense does this EO have anything to do with privacy? It is a more limited version of Lieberman-Collins; you can read that bill online right now, and it's linked from the story. You should be able to find a clause or pattern of clauses that points to "erosion of privacy".
I don't know if the update that added it was present when you read it, but the quote from TechDirt explains why it might be a "privacy disaster."
And the post on OpenCongress mentions that "inclusion of this report [to examine the legality of limiting liability of private actors for disclosing information] suggests that the Administration may believe there is a potential work-around for the privacy laws."