The Fair Use Doctrine allows someone to share copyrighted material for a limited and "transformative" purpose. Typically "transformative" falls into one of two buckets: parody or commentary/criticism. Reddit, Twitter, Facebook and Instagram all display the entire news article within their app
only after it has been presented alongside the poster's commentary. OP's Hacker News Zero app does it the other way around. Click on the link and you're taken directly to the news article and, if you then choose, you can select the "comments" to view commentary/criticism. The initial viewing of the article doesn't fall within Fair Use because it's not transformative in any way.
I suspect if the app is changed to operate more like Reddit (view the article title/source and view comments, if any, prior to opening the article) it would be approved by Apple. Alternatively, as suggested by Apple, the app could initially present only an excerpt of the article and fall within the exception but, as pointed out in the medium post, that probably isn't particularly valuable to HN users.
You also could have tried reading the link I included with my original comment.